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The comparison of any physical quantity with its standard unit is called measurement.
Physical Quantities All the quantities in terms of which laws of physics are described, and whose measurement is necessary are called physical quantities.
Units A definite amount of a physical quantity is taken as its standard unit. The standard unit should be easily reproducible, internationally accepted.
Fundamental Units Those physical quantities which are independent to each other are called fundamental quantities and their units are called fundamental units.
Derived Units Those physical quantities which are derived from fundamental quantities are called derived quantities and their units are called derived units. e.g., velocity, acceleration, force, work etc.
Definitions of Fundamental Units
The seven fundamental units of SI have been defined as under.
1. 1 kilogram A cylindrical prototype mass made of platinum and iridium alloys of height 39 mm and diameter 39 mm. It is mass of 5.0188 x 10 atoms of carbon-12.
2. 1 metre 1 metre is the distance that contains 1650763.73 wavelength of orange-red light of Kr-86. 3. 1 second 1 second is the time in which cesium atom vibrates 9192631770 times in an atomic clock. 4. 1 kelvin 1 kelvin is the (1/273.16) part of the thermodynamics temperature of the triple point of water.
5. 1 candela 1 candela is (1/60) luminous intensity of an ideal source by an area of cm’ when source is at melting point of platinum (1760°C).
6. 1 ampere 1 ampere is the electric current which it maintained in two straight parallel conductor of infinite length and of negligible cross-section area placed one metre apart in vacuum will produce between them a force 2 x 10 N per metre length.
7. 1 mole 1 mole is the amount of substance of a system which contains a many elementary entities (may be atoms, molecules, ions, electrons or group of particles, as this and atoms in 0.012 kg of carbon isotope C .
Systems of Units A system of units is the complete set of units, both fundamental and derived, for all kinds of physical quantities. The common system of units which is used in mechanics are given below:
1. CGS System In this system, the unit of length is centimetre, the unit of mass is gram and the unit of time is second.
2. FPS System In this system, the unit of length is foot, the unit of mass is pound and the unit of time is second.
3. MKS System In this system, the unit of length is metre, the unit of mass is kilogram and the unit of time is second. 25 -7 6 12
4. SI System This system contain seven fundamental units and two supplementary fundamental units.
Dimensions of any physical quantity are those powers which are raised on fundamental units to express its unit. The expression which shows how and which of the base quantities represent the dimensions of a physical quantity, is called the dimensional formula. Dimensional Formula of Some Physical Quantities
Homogeneity Principle If the dimensions of left hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle. Mathematically [LHS] = [RHS]
Applications of Dimensions
1. To check the accuracy of physical equations.
2. To change a physical quantity from one system of units to another system of units.
3. To obtain a relation between different physical quantities. Significant Figures In the measured value of a physical quantity, the number of digits about the correctness of which we are sure plus the next doubtful digit, are called the significant figures.
Rules for Finding Significant Figures
1. All non-zeros digits are significant figures, e.g., 4362 m has 4 significant figures.
2. All zeros occuring between non-zero digits are significant figures, e.g., 1005 has 4 significant figures.
3. All zeros to the right of the last non-zero digit are not significant, e.g., 6250 has only 3 significant figures.
4. In a digit less than one, all zeros to the right of the decimal point and to the left of a non-zero digit are not significant, e.g., 0.00325 has only 3 significant figures.
5. All zeros to the right of a non-zero digit in the decimal part are significant, e.g., 1.4750 has 5 significant figures.
Significant Figures in Algebric Operations
(i) In Addition or Subtraction In addition or subtraction of the numerical values the final result should retain the least decimal place as in the various numerical values. e.g., If l = 4.326 m and l = 1.50 m Then, l + l = (4.326 + 1.50) m = 5.826 m 1 2 1 2 As l has measured upto two decimal places, therefore l + l = 5.83 m
(ii) In Multiplication or Division In multiplication or division of the numerical values, the final result should retain the least significant figures as the various numerical values. e.g., If length 1= 12.5 m and breadth b = 4.125 m. Then, area A = l x b = 12.5 x 4.125 = 51.5625 m As l has only 3 significant figures, therefore A= 51.6 m Rules of Rounding Off Significant . If the digit to be dropped is less than 5, then the preceding digit is left unchanged. e.g., 1.54 is rounded off to 1.5. 2. If the digit to be dropped is greater than 5, then the preceding digit is raised by one. e.g., 2.49 is rounded off to 2.5. 3. If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. e.g., 3.55 is rounded off to 3.6. 4. If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. e.g., 3.750 is rounded off to 3.8 and 4.650 is rounded off to 4.6.
Error The lack in accuracy in the measurement due to the limit of accuracy of the instrument or due to any other cause is called an error.
1. Absolute Error The difference between the true value and the measured value of a quantity is called absolute error. If a , a , a ,…, a are the measured values of any quantity a in an experiment performed n times, then the arithmetic mean of these values is called the true value (a ) of the quantity. 2 1 2 2 2 1 2 3 n m The absolute error in measured values is given by Δa = a – a Δa = a – a …………. Δa = Δa – Δa
2. Mean Absolute Error The arithmetic mean of the magnitude of absolute errors in all the measurement is called mean absolute error.
3. Relative Error The ratio of mean absolute error to the true value is called relative
Vector can be divided into two types
1. Polar Vectors These are those vectors which have a starting point or a point of application as a displacement, force etc.
2. Axial Vectors These are those vectors which represent rotational effect and act along the axis of rotation in accordance with right hand screw rule as angular velocity, torque, angular momentum etc.
Scalars Those physical quantities which require only magnitude but no direction for their complete representation, are called scalars. Distance, speed, work, mass, density, etc are the examples of scalars. Scalars can be added, subtracted, multiplied or divided by simple algebraic laws.
Tensors Tensors are those physical quantities which have different values in different directions at the same point. Moment of inertia, radius of gyration, modulus of elasticity, pressure, stress, conductivity, resistivity, refractive index, wave velocity and density, etc are the examples of tensors. Magnitude of tensor is not unique.
Different Types of Vectors
(i) Equal Vectors Two vectors of equal magnitude, in same direction are called equal vectors.
(ii) Negative Vectors Two vectors of equal magnitude but in opposite directions are called negative vectors.
(iii) Zero Vector or Null Vector A vector whose magnitude is zero is known as a zero or null vector. Its direction is not defined. It is denoted by 0. Velocity of a stationary object, acceleration of an object moving with uniform velocity and resultant of two equal and opposite vectors are the examples of null vector.
(iv) Unit Vector A vector having unit magnitude is called a unit vector. A unit vector in the direction of vector A is given by Â = A / A A unit vector is unitless and dimensionless vector and represents direction only.
(v) Orthogonal Unit Vectors The unit vectors along the direction of orthogonal axis, i.e., X – axis, Y – axis and Z – axis are called orthogonal unit vectors. They are represented by
(vi) Co-initial Vectors Vectors having a common initial point, are called co-initial vectors.
(vii) Collinear Vectors Vectors having equal or unequal magnitudes but acting along the same or Ab parallel lines are called collinear vectors.
(viii) Coplanar Vectors Vectors acting in the same plane are called coplanar vectors.
(ix) Localised Vector A vector whose initial point is fixed, is called a localised vector.
(x) Non-localised or Free Vector A vector whose initial point is not fixed is called a non-localised or a free vector.
(xi) Position Vector A vector representing the straight line distance and the direction of any point or object with respect to the origin, is called position vector.
Addition of Vectors
1. Triangle Law of Vectors If two vectors acting at a point are represented in magnitude and direction by the two sides of a triangle taken in one order, then their resultant is represented by the third side of the triangle taken in the opposite order. If two vectors A and B acting at a point are inclined at an angle θ, R = √A + B + 2AB cos θ
If the resultant vector R subtends an angle β with vector A, then tan β = B sin θ / A + B cos θ 2.
Parallelogram Law of Vectors If two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram draw from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram draw from the same point. Resultant of vectors A and B is given by √A + B + 2AB cos θ If the resultant vector R subtends an angle β with vector A, then tan β = B sin θ / A + B cos θ
Polygon Law of Vectors It states that if number of vectors acting on a particle at a time are represented in magnitude and direction by the various sides of an open polygon taken in same order, their resultant vector E is represented in magnitude and direction by the closing side of polygon taken in opposite order. In fact, polygon law of vectors is the outcome of triangle law of vectors.
Properties of Vector Addition
(i) Vector addition is commutative, i.e., A + B = B + A
(ii) Vector addition is associative, i.e., 2 2 A +(B + C)= B + (C + A)= C + (A + B)
(iii) Vector addition is distributive, i.e., m (A + B) = m A + m B
Rotation of a Vector
(i) If a vector is rotated through an angle 0, which is not an integral multiple of 2 π, the vector changes.
(ii) If the frame of reference is rotated or translated, the given vector does not change. The components of the vector may, however, change. Resolution of a Vector into Rectangular Components If any vector A subtends an angle θ with x-axis, then its Horizontal component A = A cos θ Vertical component A = A sin θ Magnitude of vector A = √A + A tan θ = A / A
Direction Cosines of a Vector If any vector A subtend angles α, β and γ with x – axis, y – axis and z – axis respectively and its components along these axes are A , A and A , then cos α= A / A, cos β = A / A, cos γ = A / A and cos α + cos β + cos γ = 1
Subtraction of Vectors Subtraction of a vector B from a vector A is defined as the addition of vector -B (negative of vector B) to vector A Thus, A – B = A + (-B)
Multiplication of a Vector
1. By a Real Number When a vector A is multiplied by a real number n, then its magnitude becomes n times but direction and unit remains unchanged.
2. By a Scalar When a vector A is multiplied by a scalar S, then its magnitude becomes S times, and unit is the product of units of A and S but direction remains same as that of vector A.
Scalar or Dot Product of Two Vectors The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the smaller angle between them. It is denoted by . (dot). A * B = AB cos θ The scalar or dot product of two vectors is a scalar.
Properties of Scalar Product
(i) Scalar product is commutative, i.e., A * B= B * A
(ii) Scalar product is distributive, i.e., A * (B + C) = A * B + A * C
(iii) Scalar product of two perpendicular vectors is zero. A * B = AB cos 90° = O
(iv) Scalar product of two parallel vectors is equal to the product of their magnitudes, i.e., A * B = AB cos 0° = AB
(v) Scalar product of a vector with itself is equal to the square of its magnitude, i.e., A * A = AA cos 0° = A
(vi) Scalar product of orthogonal unit vectors and
(vii) Scalar product in cartesian coordinates = A B + A B + A B
Vector or Cross Product of Two Vectors The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. It is denoted by * (cross). A * B = AB sin θ n 2 x x y y z z The direction of unit vector n can be obtained from right hand thumb rule. If fingers of right hand are curled from A to B through smaller angle between them, then thumb will represent the direction of vector (A * B). The vector or cross product of two vectors is also a vector.
Properties of Vector Product
(i) Vector product is not commutative, i.e., A * B ≠ B * A [∴ (A * B) = — (B * A)]
(ii) Vector product is distributive, i.e., A * (B + C) = A * B + A * C
(iii) Vector product of two parallel vectors is zero, i.e., A * B = AB sin O° = 0
(iv) Vector product of any vector with itself is zero. A * A = AA sin O° = 0
(v) Vector product of orthogonal unit vectors
(vi) Vector product in cartesian coordinates
Direction of Vector Cross Product When C = A * B, the direction of C is at right angles to the plane containing the vectors A and B. The direction is determined by the right hand screw rule and right hand thumb rule.
(i) Right Hand Screw Rule Rotate a right handed screw from first vector (A) towards second vector (B). The direction in which the right handed screw moves gives the direction of vector (C).
(ii) Right Hand Thumb Rule Curl the fingers of your right hand from A to B. Then, the direction of the erect thumb will point in the direction of A * B.
Motion in a Straight Line
Motion If an object changes its position with respect to its surroundings with time, then it is called in motion.
Rest If an object does not change its position with respect to its surroundings with time, then it is called at rest. [Rest and motion are relative states. It means an object which is at rest in one frame of reference can be in motion in another frame of reference at the same time.]
Point Mass Object An object can be considered as a point mass object, if the distance travelled by it in motion is very large in comparison to its dimensions.
Types of Motion
1. One Dimensional Motion If only one out of three coordinates specifying the position of the object changes with respect to time, then the motion is called one dimensional motion. For instance, motion of a block in a straight line motion of a train along a straight track a man walking on a level and narrow road and object falling under gravity etc.
2. Two Dimensional Motion If only two out of three coordinates specifying the position of the object changes with respect to time, then the motion is called two dimensional motion. A circular motion is an instance of two dimensional motion.
3. Three Dimensional Motion If all the three coordinates specifying the position of the object changes with respect to time, then the motion is called three dimensional motion. A few instances of three dimension are flying bird, a flying kite, a flying aeroplane, the random motion of gas molecule etc.
Distance The length of the actual path traversed by an object is called the distance. It is a scalar quantity and it can never be zero or negative during the motion of an object. Its unit is metre.
Displacement The shortest distance between the initial and final positions of any object during motion is called displacement. The displacement of an object in a given time can be positive, zero or negative. It is a vector quantity. Its unit is metre. Speed The time rate of change of position of the object in any direction is called speed of the object. Speed (v) = Distance travelled (s) / Time taken (t) Its unit is m/s. It is a scalar quantity. Its dimensional formula is [M T ].
Uniform Speed If an object covers equal distances in equal intervals of time, then its speed is called uniform speed. Non-uniform or Variable Speed If an object covers unequal distances in equal intervals of time, then its speed is called non-uniform or variable speed.
Average Speed The ratio of the total distance travelled by the object to the total time taken is called average speed of the object.
Average speed = Total distanced travelled / Total time taken If a particle travels distances s , s , s , … with speeds v , v , v , …, then Average speed = s + s + s + ….. / (s / v + s / v + s / v + …..) If particle travels equal distances (s = s = s) with velocities v and v , then Average speed = 2 v v / (v + v ) If a particle travels with speeds v , v , v , …, during time intervals t , t , t ,…, then Average speed = v t + v t + v t +… / t + t + t +…. If particle travels with speeds v , and v for equal time intervals, i.e., t = t = t , then Average speed = v + v / 2 When a body travels equal distance with speeds V and V , the average speed (v) is the harmonic mean of two speeds. 2 / v = 1 / v + 1 / v Instantaneous Speed When an object is travelling with variable speed, then its speed at a given instant of time is called its instantaneous speed.
Velocity = Displacement / Time taken Its unit is m/s. Its dimensional formula is [M T ]. It is a vector quantity, as it has both, the magnitude and direction. The velocity of an object can be positive, zero and negative.
Uniform Velocity If an object undergoes equal displacements in equal intervals of time, then it is said to be moving with a uniform velocity.
Non-uniform or Variable Velocity If an object undergoes unequal displacements in equal intervals of time, then it is said to be moving with a non-uniform or variable velocity.
Relative Velocity Relative velocity of one object with respect to another object is the time rate of change of relative position of one object with respect to another object. Relative velocity of object A with respect to object B V = V – V When two objects are moving in the same direction, then When two objects are moving in opposite direction. When two objects are moving at an angle, then and tan β = v sin θ / v – v cos θ
Average Velocity The ratio of the total displacement to the total time taken is called average velocity. Average velocity = Total displacement / Total time taken
Acceleration The time rate of change of velocity is called acceleration. Acceleration (a) = Change in velocity (Δv) / Time interval (Δt) Its unit is m/s Its dimensional formula is [M LT ]. It is a vector quantity. Acceleration can be positive, zero or negative. Positive acceleration means velocity increasing with time, zero acceleration means velocity is uniform while negative acceleration (retardation) means velocity is decreasing with time.
Equations of Uniformly Accelerated Motion
If a body starts with velocity (u) and after time t its velocity changes to v, if the uniform acceleration is a and the distance travelled in time t in s, then the following relations are obtained, which are called equations of uniformly accelerated motion.
(i) v = u + at
(ii) s = ut + at
(iii) v = u + 2as
(iv) Distance travelled in nth second. S = u + a / 2(2n – 1)
If a body moves with uniform acceleration and velocity changes from u to v in a time interval, then the velocity at the mid point of its path √u + v / 2
Motion Under Gravity
If an object is falling freely (u = 0) under gravity, then equations of motion
(i) v = u + gt
(ii) h = ut + gt
(iii) V = u + 2gh
Note If an object is thrown upward then g is replaced by – g in above three equations. It thus follows that
(i) Time taken to reach maximum height t = u / g = √2h / g
(ii) Maximum height reached by the body h = u / 2g
(iii) A ball is dropped from a building of height h and it reaches after t seconds on earth. From the same building if two ball are thrown (one upwards and other downwards) with the same velocity u and they reach the earth surface after t, and t2 seconds respectively, then t = √t t
(iv) When a body is dropped freely from the top of the tower and another body is projected horizontally from the same point, both will reach the ground at the same time.
Projectile Motion and Circular Motion
Projectile Motion When any object is thrown from horizontal at an angle θ except 90°, then the path followed by it is called trajectory, the object is called projectile and its motion is called projectile motion. If any object is thrown with velocity u, making an angle θ, from horizontal, then Horizontal component of initial velocity = u cos θ. Vertical component of initial velocity = u sin θ. Horizontal component of velocity (u cos θ) remains same during the whole journey as no acceleration is acting horizontally. Vertical component of velocity (u sin θ) decreases gradually and becomes zero at highest point of the path. At highest point, the velocity of the body is u cos θ in horizontal direction and the angle between the velocity and acceleration is 90°.
Important Points & Formulae of Projectile Motion
1. At highest point, the linear momentum is mu cos θ and the kinetic energy is (1/2)m(u cos θ) .
2. The horizontal displacement of the projectile after t seconds x = (u cos θ)t
3. The vertical displacement of the projectile after t seconds y = (u sin θ) t — (1/2)gt
4. Equation of the path of projectile
5. The path of a projectile is parabolic.
6. Kinetic energy at lowest point = (1/2) mu
7. Linear momentum at lowest point = mu
8. Acceleration of projectile is constant throughout the motion and it acts vertically downwards being equal to g.
9. Angular momentum of projectile = mu cos θ x h, where h denotes the height.
10. In case of angular projection, the angle between velocity and acceleration varies from 0° < θ < 180°.
11. The maximum height occurs when the projectile covers a horizontal distance equal to half of the horizontal range, i.e., R/2.
12. When the maximum range of projectile is R, then its maximum height is R/4.
Time of flight It is defined as the total time for which the projectile remains in air. Maximum height It is defined as the maximum vertical distance covered by projectile.
Horizontal range It is defined as the maximum distance covered in horizontal distance.
(i) Horizontal range is maximum when it is thrown at an angle of 45° from the horizontal
(ii) For angle of projections and (90° – 0) the horizontal range is same. Projectile Projected from Some Heights
1. When Projectile is Projected Horizontally Initial velocity in vertical direction = 0
Time of flight T = √(2H/g)
Horizontal range x = uT = u √(2H/g)
Vertical velocity after t seconds v = gt (u = 0)
Velocity of projectile after t seconds If velocity makes an angle φ, from horizontal, then Equation of the path of the projectile
2. When Projectile Projected Downward at an Angle with Horizontal Initial velocity in horizontal direction = u cos θ Initial velocity in vertical direction = u sin θ Time of flight can be obtained from the equation, y y Horizontal range x = (u cos θ) t Vertical velocity after t seconds v = u sin θ + gt Velocity of projectile after t seconds
3. When Projectile Projected Upward at an Angle with Horizontal Initial velocity in horizontal direction = u cos θ Initial velocity in vertical direction = u sin θ Time of flight can be obtained from the equation Horizontal range x = (u cos θ)t Vertical velocity after t seconds, v = (- u sin θ) + gt y y Velocity of projectile after t seconcil
4. Projectile Motion on an Inclined Plane When any object is thrown with velocity u making an angle α from horizontal, at a plane inclined at an angle β from horizontal, then Initial velocity along the inclined plane = u cos (α – β) Initial velocity perpendicular to the inclined plane For angle of projections a and (90° – α + β), the range on inclined plane are same.
Circular Motion Circular motion is the movement of an object in a circular path.
1. Uniform Circular Motion If the magnitude of the velocity of the particle in circular motion remains constant, then it is called uniform circular motion.
2. Non-uniform Circular Motion If the magnitude of the velocity of the body in circular motion is n constant, then it is called non-uniform circular motion.
Note A special kind of circular motion is when an object rotates around itself. This can be called spinning motion.
Variables in Circular Motion
(i) Angular Displacement Angular displacement is the angle subtended by the position vector at the centre of the circular path. Angular displacement (Δθ) = (ΔS/r) where Δs is the linear displacement and r is the radius. Its unit is radian.
(ii) Angular Velocity The time rate of change of angular displacement (Δθ) is called angular velocity. Angular velocity (ω) = (Δθ/Δt) Angular velocity is a vector quantity and its unit is rad/s. Relation between linear velocity (v) and angular velocity (ω) is given by v = rω
(iii) Angular Acceleration The time rate of change of angular velocity (dω) is called angular acceleration. Its unit is rad/s and dimensional formula is [T ].
Relation between linear acceleration (a) and angular acceleration (α). a = rα where, r = radius Centripetal Acceleration In circular motion, an acceleration acts on the body, whose direction is always towards the centre of the path. This acceleration is called centripetal acceleration. Centripetal acceleration is also called radial acceleration as it acts along radius of circle. Its unit is in m/s and it is a vector quantity.
Centripetal Force It is that force which complex a body to move in a circular path. It is directed along radius of the circle towards its centre. For circular motion a centripetal force is required, which is not a new force but any force present there can act as centripetal force. where, m = mass of the body, c = linear velocity, ω = angular velocity and r = radius. Work done by the centripetal force is zero because the centripetal force and displacement are at right angles to each other. Examples of some incidents and the cause of centripetal force involved. .Incidents Force providing Centripetal Force
1 Orbital motion of planets. Gravitational force between planet and sun.
2 Orbital motion of electron. Electrostatic force between electron and necleus.
3 Turning of vehicles at turn. Frictional force acting between tyres of vehicle and road.
4 Motion of a stone in a circular path, tied with a string. Tension in the string. Kinematical Equations in Circular Motion Relations between different variables for an object executing circular motion are called kinematical equations in circular motion. where, ω = initial angular velocity, ω = final angular velocity, α = angular acceleration, θ = angular displacement and t = time.
Centrifugal Force Centrifugal force is equal and opposite to centripetal force. Under centrifugal force, body moves only along a straight line. It appears when centripetal force ceases to exist. 0 Centrifugal force does not act on the body in an inertial frame but arises as pseudo forces in non-inertial frames and need to be considered.
Turning at Roads If centripetal force is obtained only by the force of friction between the tyres of the vehicle and road, then for a safe turn, the coefficient of friction (µ ) between the road and tyres should be, where, v = the velocity of the vehicle and r = radius of the circular path. If centripetal force is obtained only by the banking of roads, then the speed (a) of the vehicle for a safe turn v = √rg tan θ If speed of the vehicle is less than √rg tan θ than it will move inward (down) and r will decrease and if speed is more than √rg tan θ, then it will move outward (up) and r will increase. In normal life, the centripetal force is obtained by the friction force between the road and tyres as well as by the banking of the roads. Therefore, the maximum permissible speed for the vehicle is much greater than the optimum value of the speed on a banked road. When centripetal force is obtained from friction force as well as banking of roads, then maximum safe value of speed of vehicle s When a cyclist takes turn at road, he inclined himself from the vertical, slower down his speed and move on a circular path of larger radius. If a cyclist inclined at an angle θ, then tan θ = (v /rg) where, v = speed of the cyclist, r = radius of path and g = acceleration due to gravity.
Motion in a Vertical Circle
(i) Minimum value of velocity at the highest point is √gr
(ii) The minimum velocity at the bottom required to complete the circle v = √5gr
(iii) Velocity of the body when string is in horizontal position v = √3gr
(iv) Tension in the string At the top T = 0, At the bottom T = 6 mg When string is horizontal T = 3 mg
(v) When a vehicle is moving over a convex bridge, then at the maximum height, reaction (N ) is N = mg – (mv /r) 2
(vi) When a vehicle is moving over a concave bridge, then at the lowest point, reaction (N ) is N = mg + (mv /r)
(vii) When a car takes a turn, sometimes it overturns. During the overturning, it is the inner wheel which leaves the ground first.
(viii) A driver sees a child in front of him during driving a car, then it, better to apply brake suddenly rather than taking a sharp turn to avoid an accident. Non-uniform Horizontal Circular Motion In non-uniform horizontal circular motion, the magnitude of the velocity of the body changes with time. In this condition, centripetal (radial) acceleration (a ) acts towards centre and a tangential acceleration (a ) acts towards tangent. Both acceleration acts perpendicular to each other. Resultant acceleration
where, α is angular acceleration, r = radius and a = velocity.
Conical Pendulum It consists of a string OA whose upper end 0 is fixed and bob is tied at the other free end. The string traces the surface of the cone, the arrangement is called a conical pendulum.
Laws of Motion
The property of an object by virtue of which it cannot change its state of rest or of uniform motion along a straight line its own, is called inertia. Inertia is a measure of mass of a body. Greater the mass of a body greater will be its inertia or vice-versa.
Inertia is of three types:
(i) Inertia of Rest When a bus or train starts to move suddenly, the passengers sitting in it falls backward due to inertia of rest.
(ii) Inertia of Motion When a moving bus or train stops suddenly, the passengers sitting in it jerks in forward direction due to inertia of motion.
(iii) Inertia of Direction We can protect yourself from rain by an umbrella because rain drops can not change its direction its own due to inertia of direction.
Force Force is a push or pull which changes or tries to change the state of rest, the state of uniform motion, size or shape of a body. Its SI unit is newton (N) and its dimensional formula is [MLT ].
Forces can be categorized into two types:
(i) Contact Forces Frictional force, tensional force, spring force, normal force, etc are the contact forces.
(ii) Action at a Distance Forces Electrostatic force, gravitational force, magnetic force, etc are action at a distance forces.
Impulsive Force A force which acts on body for a short interval of time, and produces a large change in momentum is called an impulsive force.
Linear Momentum The total amount of motion present in a body is called its momentum. Linear momentum of a body is equal -2 to the product of its mass and velocity. It is denoted by p. Linear momentum p = mu. Its S1 unit is kg-m/s and dimensional formula is [MLT ]. It is a vector quantity and its direction is in the direction of velocity of the body. Impulse The product of impulsive force and time for which it acts is called impulse. Impulse = Force * Time = Change in momentum Its S1 unit is newton-second or kg-m/s and its dimension is [MLT ]. It is a vector quantity and its direction is in the direction of force.
Newton’s Laws of Motion
1. Newton’s First Law of Motion A body continues to be in its state of rest or in uniform motion along a straight line unless an external force is applied on it. This law is also called law of inertia. Examples (i) When a carpet or a blanket is beaten with a stick then the dust particles separate out from it. (ii) If a moving vehicle suddenly stops then the passengers inside the vehicle bend outward.
2. Newton’s Second Law of Motion The rate of change of linear momentum is proportional to the applied force and change in momentum takes place in the direction of applied force. -1 -1 Mathematically F &infi; dp / dt F = k (d / dt) (mv) where, k is a constant of proportionality and its value is one in SI and CGS system. F= mdv / dt = ma Examples (i) It is easier for a strong adult to push a full shopping cart than it is for a baby to push the same cart. (This is depending on the net force acting on the object). (ii) It is easier for a person to push an empty shopping cart than a full one (This is depending on the mass of the object).
3. Newton’s Third Law of Motion For every action there is an equal and opposite reaction and both acts on two different bodies Mathematically F = – F Examples (i) Swimming becomes possible because of third law of motion. (ii) Jumping of a man from a boat onto the bank of a river. (iii) Jerk is produced in a gun when bullet is fired from it. (iv) Pulling of cart by a horse.
Note Newton’s second law of motion is called real law of motion because first and third laws of motion can be obtained by it.
The modern version of these laws is
(i) A body continues in its initial state of rest or motion with uniform velocity unless acted on by an unbalanced external force.
(ii) Forces always occur in pairs. If body A exerts a force on body B, an equal but opposite force is exerted by body B on body A.
Law of Conservation of Linear Momentum If no external force acts on a system, then its total linear momentum remains conserved. Linear momentum depends on frame of reference but law of conservation of linear momentum is independent of frame of reference. Newton’s laws of motion are valid only in inertial frame of reference.
Weight (w) It is a field force, the force with which a body is pulled towards the centre of the earth due to gravity. It has the magnitude mg, where m is the mass of the body and g is the acceleration due to gravity. w = mg
Rocket Rocket is an example of variable mass following law of conservation of momentum. Thrust on the rocket at any instant F = – u (dM / dt) where u = exhaust speed of the burnt and dM / dt = rate 0f gases combustion of fuel. If effect of gravity is taken into account then speed of rocket u = v + u log (M / M) – gt
Friction A force acting on the point of contact of the objects, which opposes the relative motion is called friction. It acts parallel to the contact surfaces. Frictional forces are produced due to intermolecular interactions acting between the molecules of the bodies in contact.
Friction is of three types:
1. Static Friction It is an opposing force which comes into play when one body tends to move over the surface of the other body but actual motion is not taking place. Static friction is a self adjusting force which increases as the applied force is increased,
2. Limiting Friction It is the maximum value of static friction when body is at the verge of starting motion. Limiting friction (f ) = μ R where μ , = coefficient of limiting friction and R = normal reaction. Limiting friction do not depend on area of contact surfaces but depends on their nature, i.e., smoothness or roughness. If angle of friction is θ, then coefficient of limiting friction μ = tan θ
3. Kinetic Friction If the body begins to slide on the surface, the magnitude of the frictional force rapidly decreases to a constant value f kinetic friction. Kinetic friction, f = μ N where μ k = coefficient of kinetic friction and N = normal force.
Kinetic friction is of two types:
(a) Sliding friction
(b) Rolling friction
As, rolling friction < sliding friction, therefore it is easier to roll a body than to slide. Kinetic friction (f ) = μ R where μ = coefficient of kinetic friction and R = normal reaction. Angle of repose or angle of sliding It is the minimum angle of inclination of a plane with the horizontal, such that a body placed on it, just begins to slide down. If angle of repose is a. and coefficient of limiting friction is μ, then μ = tan α
Work, Power and Energy
When a force acts on an object and the object actually moves in the direction of force, then the work is said to be done by the force. Work done by the force is equal to the product of the force and the displacement of the object in the direction of force. If under a constant force F the object displaced through a distance s, then work done by the force W = F * s = F s cos θ where a is the smaller angle between F and s. Work is a scalar quantity, Its S1 unit is joule and CGS unit is erg.
∴ 1 joule = 10 erg
Its dimensional formula is [ML T ]. Work done by a force is zero, if
(a) body is not displaced actually, i.e., s = 0
(b) body is displaced perpendicular to the direction of force, i.e., θ = 90°
Work done by a force is positive if angle between F and s is acute angle. Work done by a force is negative if angle between F and s is obtuse angle.
Work done by a constant force depends only on the initial and final Positions and not on the actual path followed between initial and final positions.
Work done in different conditions
(i) Work done by a variable force is given by W = ∫ F * ds
It is equal to the area under the force-displacement graph along with proper sign.
(ii) Work done in displacing any body under the action of a number of forces is equal to the work done by the resultant force.
(iii) In equilibrium (static or dynamic), the resultant force is zero therefore resultant work done is zero.
(iv) If work done by a force during a rough trip of a system is zero, then the force is conservative, otherwise it is called non-conservative force. Gravitational force, electrostatic force, magnetic force, etc are conservative forces. All the central forces are conservative forces. Frictional force, viscous force, etc are non-conservative forces.
(v) Work done by the force of gravity on a particle of mass m is given by W = mgh where g is acceleration due to gravity and h is height through particle one displaced.
(vi) Work done in compressing or stretching a spring is given by W = 1 / 2 kx where k is spring constant and x is displacement from mean position.
Power = Rate of doing work = Work done / Time taken If under a constant force F a body is displaced through a distance s in time t, the power p = W / t = F * s / t But s / t = v ; uniform velocity with which body is displaced.
∴ P = F * v = F v cos θ where θ is the smaller angle between F and v. power is a scalar quantity. Its S1 unit is watt and its dimensional formula is [ML T ]. Its other units are kilowatt and horse power, 1 kilowatt = 1000 watt 1 horse power = 746 watt Energy Energy of a body is its capacity of doing work. It is a scalar quantity. Its S1 unit is joule and CGS unit is erg. Its dimensional formula is [ML T ].
There are several types of energies, such as mechanical energy (kinetic energy and potential energy), chemical energy, light energy, heat energy, sound energy, nuclear energy, electric energy etc.
Mechanical Energy The sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depend upon time. This is known as law of conservation of mechanical energy.
Mechanical energy is of two types:
1. Kinetic Energy The energy possessed by any object by virtue of its motion is called its kinetic energy. Kinetic energy of an object is given by k = 1 / 2 mv = p / 2m where m = mass of the object, U = velocity of the object and p = mv = momentum of the object.
2. Potential Energy The energy possessed by any object by virtue of its position or configuration is called its potential energy.
There are three important types of potential energies:
(i) Gravitational Potential Energy If a body of mass m is raised through a height h against gravity, then its gravitational potential energy = mgh,
(ii) Elastic Potential Energy If a spring of spring constant k is stretched through a distance x. then elastic potential energy of the spring = 1 . 2 kx The variation of potential energy with distance is shown in figure. Potential energy is defined only for conservative forces. It does not exist for non-conservative forces. 2 2 2 Potential energy depends upon frame of reference.
(iii) Electric Potential Energy The electric potential energy of two point charges ql and q’l. separated by a distance r in vacuum is given by U = 1 / 4πΣ * q q / r Here 1 / 4πΣ = 9.0 * 10 N-m / C constant.
Work-Energy Theorem Work done by a force in displacing a body is equal to change in its kinetic energy. where, K = initial kinetic energy and K = final kinetic energy. Regarding the work-energy theorem it is worth noting that (i) If W is positive, then K – K = positive, i.e., K > K or kinetic energy will increase and vice-versa. (ii) This theorem can be applied to non-inertial frames also. In a non-inertial frame it can be written as: Work done by all the forces (including the Pseudo force) = change in kinetic energy in non-inertial frame. Mass-Energy Equivalence According to Einstein, the mass can be transformed into energy and vice – versa. When Δm. mass disappears, then produced energy E = Δmc where c is the speed of light in vacuum.
Principle of Conservation of Energy The sum of all kinds of energies in an isolated system remains constant at all times. Principle of Conservation of Mechanical Energy For conservative forces the sum of kinetic and potential energies of any object remains constant throughout the motion. According to the quantum physics, mass and energy are not conserved separately but are conserved as a single entity called ‘mass-energy’.
Collisions Collision between two or more particles is the interaction for a short interval of time in which they apply relatively strong forces on each other. In a collision physical contact of two bodies is not necessary.
there are two types of collisions:
1. Elastic collision The collision in which both the momentum and the kinetic energy of the system remains conserved are called elastic collisions. In an elastic collision all the involved forces are conservative forces. Total energy remains conserved.
2. Inelastic collision The collision in which only the momentum remains conserved but kinetic energy does not remain conserved are called inelastic collisions. In an inelastic collision some or all the involved forces are non-conservative forces. Total energy of the system remains conserved. If after the collision two bodies stick to each other, then the collision is said to be perfectly inelastic.
Coefficient of Restitution or Resilience The ratio of relative velocity of separation after collision to the velocity of approach before collision is called coefficient of restitution resilience. It is represented by e and it depends upon the material of the collidingI bodies. For a perfectly elastic collision, e = 1 For a perfectly inelastic collision, e = 0 For all other collisions, 0 < e < 1 One Dimensional or Head-on Collision If the initial and final velocities of colliding bodies lie along the same line, then the collision is called one dimensional or head-on collision. Inelastic One Dimensional Collision Applying Newton’s experimental law, we have Velocities after collision v = (m – m ) u + 2m u / (m + m ) and v = (m – m ) u + 2m u / (m + m ) When masses of two colliding bodies are equal, then after the collision, the bodies exchange their velocities. v = u and v = u If second body of same mass (m = m ) is at rest, then after collision first body comes to rest and second body starts moving with the initial velocity of first body. v = 0 and v = u If a light body of mass m collides with a very heavy body of mass m at rest, then after collision. v = – u and v = 0 It means light body will rebound with its own velocity and heavy body will continue to be at rest. If a very heavy body of mass m collides with a light body of mass m (m > > m ) at rest, then after collision
For rigid bodies, centre of mass is independent of the state of the body i.e., whether it is in rest or in accelerated motion centre of mass will rermain same.
Translational Motion A rigid body performs a pure translational motion, if each particle the body undergoes the same displacement in the same direction in given interval of time.
Rotational Motion A rigid body performs a pure rotational motion, if each particle of the body moves in a circle, and the centre of all the circles lie on a straight line called the axes of rotation.
Rigid Body If the relative distance between the particles of a system do not changes on applying force, then it called a rigtd body. General motion of a rigid body consists of both the translational motion and the rotational motion.
Moment of Inertia The inertia of rotational motion is called moment of inertia. It is denoted by L. Moment of inertia is the property of an object by virtue of which it opposes any change in its state of rotation about an axis. The moment of inertia of a body about a given axis is equal to the sum of the products of the masses of its constituent particles and the square of their respective distances from the axis of rotation. Its unit is kg.m and its dimensional formula is [ML ]. The moment of inertia of a body depends upon position of the axis of rotation orientation of the axis of rotation shape and size of the body distribution of mass of the body about the axis of rotation. The physical significance of the moment of inertia is same in rotational motion as the mass in linear motion.
The Radius of Gyration The root mean square distance of its constituent particles from the axis of rotation is called the radius of gyration of a body. It is denoted by K. Radius of gyration 2 2 The product of the mass of the body (M) and square of its radius gyration (K) gives the same moment of inertia of the body about rotational axis. Therefore, moment of inertia I = MK ⇒ K = √1/M
Parallel Axes Theorem The moment of inertia of any object about any arbitrary axes is equal to the sum of moment of inertia about a parallel axis passing through the centre of mass and the product of mass of the body and the square of the perpendicular distance between the two axes. Mathematically I = I + Mr where I is the moment of inertia about the arbitrary axis, I is moment of inertia about the parallel axis through the centre of mass, M is the total mass of the object and r is the perpendicular distance between the axis.
Perpendicular Axes Theorem The moment of inertia of any two dimensional body about an axis perpendicular to its plane (I ) is equal to the sum of moments of inertia of the body about two mutually perpendicular axes lying in its own plane and intersecting each other at a point, where the perpendicular axis passes through it. Mathematically I = I + I where I and I are the moments of inertia of plane lamina about perpendicular axes X and Y respectively which lie in the plane lamina an intersect each other.
Theorem of parallel axes is applicable for any type of rigid body whether it is a two dimensional or three dimensional, while the theorem of perpendicular is applicable for laminar type or two I dimensional bodies only.
Moment of Inertia of Homogeneous Rigid Bodies
Equations of Rotational Motion
(i) ω = ω + αt
(ii) θ = ω t + 1/2 αt
(iii) ω = ω + 2αθ
where θ is displacement in rotational motion, ω is initial velocity, omega; is final velocity and a is acceleration.
Torque Torque or moment of a force about the axis of rotation τ = r x F = rF sinθ n It is a vector quantity. If the nature of the force is to rotate the object clockwise, then torque is called negative and if rotate the object anticlockwise, then it is called positive. Its SI unit is ‘newton-metre’ and its dimension is [ML T ]. In rotational motion, torque, τ = Iα where a is angular acceleration and 1is moment of inertia.
Angular Momentum The moment of linear momentum is called angular momentum. It is denoted by L. Angular momentum, L = I ω = mvr In vector form, L = I ω = r x mv Its unit is ‘joule-second’ and its dimensional formula is [ML T ]. Torque, τ = dL/dt
Conservation of Angular Momentum If the external torque acting on a system is zero, then its angular momentum remains conserved. If τ 0, then L = I(ω) = constant ⇒ I ω == I ω Angular
Impulse Total effect of a torque applied on a rotating body in a given time is called angular impulse. Angular impulse is equal to total change in angular momentum of the system in given time. Thus, angular impulse
Every object in the universe attracts every other object with a force which is called the force of gravitation. Gravitation is one of the four classes of interactions found in nature. These are (i) the gravitational force (ii) the electromagnetic force (iii) the strong nuclear force (also called the hadronic force). (iv) the weak nuclear forces. Although, of negligible importance in the interactions of elementary particles, gravity is of primary importance in the interactions of objects. It is gravity that holds the universe together.
Newton’s Law of Gravitation Gravitational force is a attractive force between two masses m and m separated by a distance r. The gravitational force acting between two point objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. Gravitational force. where G is universal gravitational constant. The value of G is 6.67 X 10 Nm kg and is same throughout the universe. The value of G is independent of the nature and size of the bodies well as the nature of the medium between them. Dimensional formula of Gis [M L T ].
Important Points about Gravitation Force
(i) Gravitational force is a central as well as conservative force.
(ii) It is the weakest force in nature.
(iii) It is 1036 times smaller than electrostatic force and 10’l8times smaller than nuclear force.
(iv) The law of gravitational is applicable for all bodies, irrespective of their size, shape and position. (v) Gravitational force acting between sun and planet provide it centripetal force for orbital motion. (vi) Gravitational pull of the earth is called gravity.
(vii) Newton’s third law of motion holds good for the force of gravitation. It means the gravitation forces between two bodies are action-reaction pairs.
Following three points are important regarding the gravitational force
(i) Unlike the electrostatic force, it is independent of the medium between the particles.
(ii) It is conservative in nature.
(iii) It expresses the force between two point masses (of negligible volume). However, for external points of spherical bodies the whole mass can be assumed to be concentrated at its centre of mass.
Note Newton’s law of gravitation holde goods for object lying at uery large distances and also at very short distances. It fails when the distance between the objects is less than 10-9 m i.e., of the order of intermolecular distances.
Acceleration Due to Gravity The uniform acceleration produced in a freely falling object due to the gravitational pull of the earth is known as acceleration due to gravity. It is denoted by g and its unit is m/s . It is a vector quantity and its direction is towards the centre of the earth. The value of g is independent of the mass of the object which is falling freely under gravity. The value of g changes slightly from place to place. The value of g is taken to be 9.8 m/s for all practical purposes. The value of acceleration due to gravity on the moon is about. one sixth of that On the earth and on the sun 2 2 is about 27 times of that on the earth. Among the planets, the acceleration due to gravity is minimum on the mercury. Relation between g and a is given by g = Gm / R where M = mass of the earth = 6.0 * 10 kg and R = radius of the earth = 6.38 * 10 m. Acceleration due to gravity at a height h above the surface of the earth is given by g = Gm / (R+h) = g (1 – 2h / R)
Factors Affecting Acceleration Due to Gravity
(i) Shape of Earth Acceleration due to gravity g &infi; 1 / R Earth is elliptical in shape. Its diameter at poles is approximately 42 km less than its diameter at equator. Therefore, g is minimum at equator and maximum at poles.
(ii) Rotation of Earth about Its Own Axis If ω is the angular velocity of rotation of earth about its own axis, then acceleration due to gravity at a place having latitude λ is given by g’ = g – Rω cos λ At poles λ = 90° and g’ = g Therefore, there is no effect of rotation of earth about its own axis at poles. At equator λ = 0° and g’ = g – Rω The value of g is minimum at equator If earth stapes its rotation about its own axis, then g will remain unchanged at poles but increases by Rω at equator. (iii) Effect of Altitude The value of g at height h from earth’s surface g’ = g / (1 + h / R) Therefore g decreases with altitude.
(iv) Effect of Depth The value of gat depth h A from earth’s surface g’ = g * (1 – h / R) Therefore g decreases with depth from earth’s surface. The value of g becomes zero at earth’s centre.
Gravitational Field The space in the surrounding of any body in which its gravitational pull can be experienced by other bodies is called gravitational field. Intensity of Gravitational Field The gravitational force acting per unit mass at Earth any point in gravitational field is called intensity of gravitational field at that point. It is denoted by E or I. E or I = F / m Intensity of gravitational field at a distance r from a body of mass M is given by E or I = GM / r It is a vector quantity and its direction is towards the centre of gravity of the body. Its S1 unit is N/m and its dimensional formula is [LT ].
Gravitational mass M is defined by Newton’s law of gravitation. M = F / g = W / g = Weight of body / Acceleration due to gravity ∴ (M )g / (M )g = F / F g g g 2 -2 g g g 1 2 g1g2 g2g1 Gravitational Potential Gravitational potential at any point in gravitational field is equal the work done per unit mass in bringing a very light body from infinity to that point. It is denoted by V . Gravitational potential, V = W / m = – GM / r Its SI unit is J / kg and it is a scalar quantity. Its dimensional formula is [L r ]. Since work W is obtained, that is, it is negative, the gravitational potential is always negative.
Gravitational Potential Energy Gravitational potential energy of any object at any point in gravitational field is equal to the work done in bringing it from infinity to that point. It is denoted by U. Gravitational potential energy U = – GMm / r The negative sign shows that the gravitational potential energy decreases with increase in distance. Gravitational potential energy at height h from surface of earth U = – GMm / R + h = mgR / 1 + h/R g
Satellite A heavenly object which revolves around a planet is called a satellite. Natural satellites are those heavenly objects which are not man made and revolve around the earth.
Artificial satellites are those neaven objects which are man made and launched for some purposes revolve around the earth. Time period of satellite T = 2π √r / GM = 2π √(R + h) / g [ g = GM / R Near the earth surface, time period of the satellite T = 2π √R / GM = √3π / Gp T = 2π √R / g = 5.08 * 10 s = 84 min. where p is the average density of earth.
Artificial satellites are of two types :
1. Geostationary or Parking Satellites A satellite which appears to be at a fixed position at a definite height to an observer on earth is called geostationary or parking satellite. Height from earth’s surface = 36000 km Radius of orbit = 42400 km Time period = 24 h Orbital velocity = 3.1 km/s Angular velocity = 2π / 24 = π / 12 rad / h There satellites revolve around the earth in equatorial orbits. The angular velocity of the satellite is same in magnitude and direction as that of angular velocity of the earth about its own axis. These satellites are used in communication purpose. INSAT 2B and INSAT 2C are geostationary satellites of India.
2. Polar Satellites These are those satellites which revolve in polar orbits around earth. A polar orbit is that orbit whose angle of inclination with equatorial plane of earth is 90°. Height from earth’s surface = 880 km Time period = 84 min Orbital velocity = 8 km / s Angular velocity = 2π / 84 = π / 42 rad / min. There satellites revolve around the earth in polar orbits. These satellites are used in forecasting weather, studying the upper region of the atmosphere, in mapping, etc. PSLV series satellites are polar satellites of India.
Orbital Velocity Orbital velocity of a satellite is the minimum velocity required to the satellite into a given orbit around earth. Orbital velocity of a satellite is given by v = √GM / r = R √g / R + h where, M = mass of the planet, R = radius of the planet and h = height of the satellite from planet’s surface. If satellite is revolving near the earth’s surface, then r = (R + h) =- R Now orbital velocity, v = √gR = 7.92km / h if v is the speed of a satellite in its orbit and v is the required orbital velocity to move in the orbit, then (i) If v < v , then satellite will move on a parabolic path and satellite falls back to earth. (ii) If V = v then satellite revolves in circular path/orbit around earth. (iii) If v < V < v then satellite shall revolve around earth in elliptical orbit. Energy of a Satellite in Orbit Total energy of a satellite E = KE + PE = GMm / 2r + (- GMm / r) = – GMm / 2r
Binding Energy The energy required to remove a satellite from its orbit around the earth (planet) to infinity is called binding energy of the satellite. Binding energy of the satellite of mass m is given by BE = + GMm / 2r Escape Velocity Escape velocity on earth is the minimum velocity with which a body has to be projected vertically upwards from the earth’s surface so that it just crosses the earth’s gravitational field and never returns. Escape velocity of any object v = √2GM / R = √2gR = √8πp GR / 3 Escape velocity does not depend upon the mass or shape or size of the body as well as the direction of projection of the body. Escape velocity at earth is 11.2 km / s. Some Important Escape Velocities Heavenly body Escape velocity Moon 2.3 km/s Mercury 4.28 km/s Earth 11.2 km/s Jupiter 60 km/s Sun 618 km/s e 2 Neutron star 2 x 10 km/s Relation between escape velocity and orbital velocity of the satellite v = √2 v If velocity of projection U is equal the escape velocity (v = v ), then the satellite will escape away following a parabolic path. If velocity of projection u of satellite is greater than the escape velocity ( v > v ), then the satellite will escape away following a hyperbolic path.
Weightlessness It is a situation in which the effective weight of the body becomes zero, Weightlessness is achieved
(i) during freely falling under gravity
(ii) inside a space craft or satellite
(iii) at the centre of the earth
(iv) when a body is lying in a freely falling lift.
Kepler’s Laws of Planetary Motion
(i) Law of orbit Every planet revolve around the sun in elliptical orbit and sun is at its one focus.
(ii) Law of area The radius vector drawn from the sun to a planet sweeps out equal areas in equal intervals of time, i.e., the areal velocity of the planet around the sun is constant. Areal velocity of a planet dA / dt = L / 2m = constant where L = angular momentum and m = mass of the planet.
(iii) Law of period The square of the time period of revolution of planet around the sun is directly proportional to the cube semi-major axis of its elliptical orbit. T &infi; a or (T / T ) = (a / a ) where, a = semi-major axis of the elliptical orbit.
(i) A missile is launched with a velocity less than the escape velocity. The sum of its kinetic energy and potential energy is negative.
(ii) The orbital speed of jupiter is less than the orbital speed of earth.
(iii) A bomb explodes on the moon. You cannot hear the sound of the explosion on earth.
(iv) A bottle filled with water at 30°C and fitted with a cork is taken to the moon. If the cork is opened at the surface of the moon then water will boil.
(v) For a satellite orbiting near earth’s surface (a) Orbital velocity = 8 km / s (b) Time period = 84 min approximately (c) Angular speed ω = 2π / 84 rad / min = 0.00125 rad / s
(vi) Inertial mass and gravitational mass (a) Inertial mass = force / acceleration (b) Gravitational mass = weight of body / acceleration due to gravity (c) They are equal to each other in magnitude. (d) Gravitational mass of a body is affected by the presence of other bodies near it. Inertial mass of a body remains unaffected by the presence of other bodies near it.
Deforming Force A force which produces a change in configuration of the object on applying it, is called a deforming force.
Elasticity Elasticity is that property of the object by virtue of which it regain its original configuration after the removal of the deforming force.
Elastic Limit Elastic limit is the upper limit of deforming force upto which, if deforming force is removed, the body regains its original form completely and beyond which if deforming force is increased the body loses its property of elasticity and get permanently deformed.
Perfectly Elastic Bodies Those bodies which regain its original configuration immediately and completely after the removal of deforming force are called perfectly elastic bodies. e.g., quartz and phosphor bronze etc.
Perfectly Plastic Bodies Those bodies which does not regain its original configuration at all on the removal of deforming force are called perfectly plastic bodies, e.g., putty, paraffin, wax etc.
Stress The internal restoring force acting per unit area of a deformed body is called stress. Stress = Restoring force / Area Its unit is N/m or Pascal and dimensional formula is [ML T ]. Stress is a tensor quantity.
Stress is of Two Types
(i) Normal Stress If deforming force is applied normal to the area, then the stress is called normal stress. If there is an increase in length, then stress is called tensile stress. If there is a decrease in length, then stress is called compression stress.
(ii) Tangential Stress If deforming force is applied tangentially, then the stress is called tangential stress.
Strain The fractional change in configuration is called strain. Strain = Change in the configuration / Original configuration It has no unit and it is a dimensionless quantity.
According to the change in configuration, the strain is of three types
(1) Longitudinal strain= Change in length / Original length
(2) Volumetric strain = Change in volume / Original volume
(iii) Shearing strain = Angular displacement of the plane perpendicular to the fixed surface.
Hooke’s Law Within the limit of elasticity, the stress is proportional to the strain. Stress &infi; Strain or Stress = E * Strain where, E is the modulus of elasticity of the material of the body.
Types of Modulus of Elasticity
1. Young’s Modulus of Elasticity It is defined as the ratio of normal stress to the longitudinal strain Within the elastic limit. y = Normal stress / Longitudinal strain y = FΔl / Al = Mg Δl / πr Its unit is N/m or Pascal and its dimensional formula is [ML T ].
2. Bulk Modulus of Elasticity It is defined as the ratio of normal stress to the volumetric strain within the elastic limit. K = Normal stress / Volumetric strain K = FV / A ΔV = &DElta;p V / Δ V where, Δp = F / A = Change in pressure. Its unit is N/m or Pascal and its dimensional formula is [ML T ].
3. Modulus of Rigidity (η) It is defined as the ratio of tangential stress to the shearing strain, within the elastic limit. η = Tangential stress / Shearing strain Its urut is N/m or Pascal and its dimensional formula is [ML T ].
Compressibility Compressibility of a material is the reciprocal of its bulk modulus of elasticity. Compressibility (C) = 1 / k Its SI unit is N m and CGS unit is dyne cm . Steel is more elastic than rubber. Solids are more elastic and gases are least elastic. For liquids. modulus of rigidity is zero. Young’s modulus (Y) and modulus of rigidity (η) are possessed by solid materials only. Limit of Elasticity The maximum value of deforming force for which elasticity is present in the body is called its limit of elasticity.
Breaking Stress The minimum value of stress required to break a wire, is called breaking stress. Breaking stress is fixed for a material but breaking force varies with area of cross-section of the wire. Safety factor = Breaking stress / Working stress Elastic Relaxation Time The time delay in restoring the original configuration after removal of deforming force is called elastic relaxation time. For quartz and phosphor bronze this time is negligible.
Elastic After Effect The temporary delay in regaining the original configuration by the elastic body after the removal of deforming force, is called elastic after effect.
Elastic Fatigue The property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming force is called elastic fatigue.
Ductile Materials The materials which show large plastic range beyond elastic limit are called ductile materials, e.g., copper, silver, iron, aluminum, etc. Ductile materials are used for making springs and sheets.
Brittle Materials The materials which show very small plastic range beyond elastic limit are called brittle materials, e.g., glass, cast iron, etc.
Elastomers The materials for which strain produced is much larger than the stress applied, with in the limit of elasticity are called elastomers, e.g., rubber, the elastic tissue of aorta, the large vessel carrying blood from heart. etc. Elastomers have no plastic range.
Elastic Potential Energy in a Stretched Wire The work done in stretching a wire is stored in form of potential energy of the wire. Potential energy U = Average force * Increase in length = 1 / 2 FΔl = 1 / 2 Stress * Strain * Volume of the wire Elastic potential energy per unit volume U = 1 / 2 * Stress * Strain = 1 / 2 (Young’s modulus) * (Strain) Elastic potential energy of a stretched spring = 1 / 2 kx where, k = Force constant of spring and x = Change in length. Thermal Stress When temperature of a rod fixed at its both ends is changed, then the produced stress is called thermal stress. Thermal stress = F / A = yαΔθ where, α = coefficient of linear expansion of the material of the rod. When temperature of a gas enclosed in a vessel is changed, then the thermal stress produced is equal to change in pressure (Δp)of the gas. 2 2 Thermal stress = Δ p = Ky Δ θ where, K = bulk modulus of elasticity and γ = coefficient of cubical expansion of the gas. Interatomic force constant K = Yr where, r = interatomic distance.
Poisson’s Ratio When a deforming force is applied at the free end of a suspended wire of length 1 and radius R, then its length increases by dl but its radius decreases by dR. Now two types of strains are produced by a single force.
(i) Longitudinal strain = &DElta;
(ii) Lateral strain = – Δ R/ R
∴ Poisson’s Ratio (σ) = Lateral strain / Longitudinal strain = – Δ R/ R / ΔU
The theoretical value of Poisson’s ratio lies between – 1 and 0.5. Its practical value lies between 0 and 0.5.
Relation Between Y, K, η and σ
(i) Y = 3K (1 – 2σ)
(ii) Y = 2 η ( 1 + σ)
(iii) σ = 3K – 2η / 2η + 6K
(iv) 9 / Y = 1 / K + 3 / η or Y = 9K η / η + 3K
Important Points Coefficient of elasticity depends upon the material, its temperature and purity but not on stress or strain. For the same material, the three coefficients of elasticity γ, η and K have different magnitudes.
Isothermal elasticity of a gas E = ρ where, ρ = pressure of the gas.
Adiabatic elasticity of a gas E = γρ where, γ = C / C ratio of specific heats at constant pressure and at constant volume.
Ratio between isothermal elasticity and adiabatic elasticity E / E = γ = C / C
Cantilever A beam clamped at one end and loaded at free end is called a cantilever. Depression at the free end of a cantilever is given by δ = wl / 3YI where, w = load, 1 = length of the cantilever, y = Young’s modulus of elasticity, and I = geometrical moment of inertia. For a beam of rectangular cross-section having breadth b and thickness d. I = bd / 12 For a beam of circular cross-section area having radius r, I = π r / 4 Beam Supported at Two Ends and Loaded at the Middle Depression at middle δ = wl / 48YI
Torsion of a Cylinder where, η = modulus of rigidity of the material of cylinder, r = radius of cylinder, and 1 = length of cylinder, Work done in twisting the cylinder through an angle θ W = 1 / 2 Cθ Relation between angle of twist (θ) and angle of shear (φ) rθ = lφ or φ = r / l = θ
Fluids Fluids are those substances which can flow when an external force is applied on it.
Liquids and gases are fluids. Fluids do not have finite shape but takes the shape of the containing vessel, The total normal force exerted by liquid at rest on a given surface is called thrust of liquid. The SI unit of thrust is newton. In fluid mechanics the following properties of fluid would be considered
(i) When the fluid is at rest – hydrostatics
(ii) When the fluid is in motion – hydrodynamics Pressure Exerted by the Liquid The normal force exerted by a liquid per unit area of the surface in contact is called pressure of liquid or hydrostatic pressure. Pressure exerted by a liquid column p = hρg Where, h = height of liquid column, ρ = density of liquid and g = acceleration due to gravity Mean pressure on the walls of a vessel containing liquid upto height h is (hρg / 2).
Pascal’s Law The increase in pressure at a point in the enclosed liquid in equilibrium is transmitted equally in all directions in liquid and to the Walls of the container. The working of hydraulic lift, hydraulic press and hydraulic brakes are based on Pascal’s law.
Atmospheric Pressure The pressure exerted by the atmosphere on earth is atmospheric pressure. It is about 100000 N/m . It is equivalent to a weight of 10 tones on 1 m . At sea level, atmospheric pressure is equal to 76 cm of mercury column. Then, atmospheric pressure = hdg = 76 x 13.6 x 980 dyne/cm [The atmospheric pressure does not crush our body because the pressure of the blood flowing through our circulatory system] balanced this pressure.] Atmospheric pressure is also measured in torr and bar. 1 torr = 1 mm of mercury column 1 bar = l0 Pa Aneroid barometer is used to measure atmospheric pressure.
Buoyancy When a body is partially or fully immersed in a fluid an upward force acts on it, which is called buoyant force or simply buoyancy. The buoyant force acts at the centre of gravity of the liquid displaced] by the immersed part of the body and this point is called the centre buoyancy.
Archimedes’ Principle When a body is partially or fully immersed in a liquid, it loses some of its weight. and it is equal to the weight of the liquid displaced by the immersed part of the body. If T is the observed weight of a body of density σ when it is fully immersed in a liquid of density p, then real weight of the body w = T / ( 1 – p / σ)
Laws of Floatation A body will float in a liquid, if the weight of the body is equal to the weight of the liquid displaced by the immersed part of the body. If W is the weight of the body and w is the buoyant force, then
(a) If W > w, then body will sink to the bottom of the liquid.
(b) IfW < w, then body will float partially submerged in the liquid.
(c) If W = w, then body will float in liquid if its whole volume is just immersed in the liquid, The floating body will be in stable equilibrium if meta-centre (centre of buoyancy) lies vertically above the centre of gravity of the body. The floating body will be in unstable equilibrium if meta-centre (centre of buoyancy) lies vertically below the centre of gravity of the body. The floating body will be in neutral equilibrium if meta-centre (centre of buoyancy) coincides with the centre of gravity of the body.
Density and Relative Density Density of a substance is defined as the ratio of its mass to its volume. Density of a liquid = Mass / Volume Density of water = 1 g/cm or l0 kg/m It is scalar quantity and its dimensional formula is [ML ].
Relative density of a substance is defined as the ratio of its density to the density of water at 4°C, Relative density = Density of substance / Density of water at 4°C = Weight of substance in air / Loss of weight in water Relative density also known as specific gravity has no unit, no dimensions. For a solid body, density of body = density of substance While for a hollow body, density of body is lesser than that of Substance. When immiscible liquids of different densities are poured in a container, the liquid of highest density will be at the bottom while, that of lowest density at the top and interfaces will be plane.
Density of a Mixture of Substances When two liquids of mass m and m having density p and p are mixed together then density of mixture is p = m + m / (m /p ) + (m + p ) = p p (m + m ) / (m p + m p ) When two liquids of same mass m but of different densities p and p are mixed together then density of mixture is p = 2p p / p + p When two liquids of same volume V but of different densities p and p are mixed together then density of mixture is p = p + p / 2
Density of a liquid varies with pressure p = p [ 1 + Δp / K] where, p = initial density of the liquid, K = bulk modulus of elasticity of the liquid and Δp = change in pressure
Thermometry and Calorimetry
The branch dealing with measurement of temperature is called thremometry and the devices used to measure temperature are called thermometers.
Heat Heat is a form of energy called thermal energy which flows from a higher temperature body to a lower temperature body when they are placed in contact. Heat or thermal energy of a body is the sum of kinetic energies of all its constituent particles, on account of translational, vibrational and rotational motion. The SI unit of heat energy is joule (J). The practical unit of heat energy is calorie. 1 cal = 4.18 J 1 calorie is the quantity of heat required to raise the temperature of 1 g of water by 1°C. Mechanical energy or work (W) can be converted into heat (Q) by 1 W = JQ where J = Joule’s mechanical equivalent of heat. J is a conversion factor (not a physical quantity) and its value is 4.186 J/cal.
Temperature Temperature of a body is the degree of hotness or coldness of the body. A device which is used to measure the temperature, is called a thermometer. Highest possible temperature achieved in laboratory is about 108 while lowest possible temperature attained is 10-8 K. Branch of Physics dealing with production and measurement temperature close to 0 K is known as cryagenics, while that deaf with the measurement of very high temperature is called pyromet Temperature of the core of the sun is 107 K while that of its surface 6000 K. NTP or STP implies 273.15 K (0°C = 32°F).
Different Scale of Temperature
1. Celsius Scale In this scale of temperature, the melting point ice is taken as 0°C and the boiling point of water as 100°C and space between these two points is divided into 100 equal parts
2. Fahrenheit Scale In this scale of temperature, the melt point of ice is taken as 32°F and the boiling point of water as 211 and the space between these two points is divided into 180 equal parts.
3. Kelvin Scale In this scale of temperature, the melting pouxl ice is taken as 273 K and the boiling point of water as 373 K the space between these two points is divided into 100 equal pss
Relation between Different Scales of Temperatures Thermometric Property The property of an object which changes with temperature, is call thermometric property. Different thermometric properties thermometers have been given below
(i) Pressure of a Gas at Constant Volume where p, p . and p , are pressure of a gas at constant volume 0°C, 100°C and t°C. A constant volume gas thermometer can measure tempera from – 200°C to 500°C.
(ii) Electrical Resistance of Metals R = R (1 + αt + βt ) where α and β are constants for a metal. As β is too small therefore we can take 100 t t 0 2 R = R (1 + αt) where, α = temperature coefficient of resistance and R and R , are electrical resistances at 0°C and t°C. where R and R are electrical resistances at temperatures t and t . where R is the resistance at 100°C. Platinum resistance thermometer can measure temperature from —200°C to 1200°C.
(iii) Length of Mercury Column in a Capillary Tube l = l (1 + αt) where α = coefficient of linear expansion and l , l are lengths of mercury column at 0°C and t°C.
Thermo Electro Motive Force When two junctions of a thermocouple are kept at different temperatures, then a thermo-emf is produced between the junctions, which changes with temperature difference between the junctions. Thermo-emf E = at + bt where a and b are constants for the pair of metals. Unknown temperature of hot junction when cold junction is at 0°C. Where E is the thermo-emf when hot junction is at 100°C. A thermo-couple thermometer can measure temperature from —200°C to 1600°C.
Thermal Equilibrium When there is no transfer of heat between two bodies in contact, the the bodies are called in thermal equilibrium. Zeroth Law of Thermodynamics If two bodies A and B are separately in thermal equilibrium with thirtli body C, then bodies A and B will be in thermal equilibrium with each other.
Triple Point of Water The values of pressure and temperature at which water coexists inequilibrium in all three states of matter, i.e., ice, water and vapour called triple point of water. Triple point of water is 273 K temperature and 0.46 cm of mere pressure. Specific Heat The amount of heat required to raise the temperature of unit mass the substance through 1°C is called its specific heat. It is denoted by c or s. Its SI unit is joule/kilogram-°C'(J/kg-°C). Its dimensions is [L T θ ]. The specific heat of water is 4200 J kg °C or 1 cal g C , which high compared with most other substances. Gases have two types of specific heat 1. The specific heat capacity at constant volume (C ). 2. The specific heat capacity at constant pressure (C ). Specific heat at constant pressure (C ) is greater than specific heat constant volume (C ), i.e., C > C . For molar specific heats C – C = R where R = gas constant and this relation is called Mayer’s formula. The ratio of two principal sepecific heats of a gas is represented by γ. The value of y depends on atomicity of the gas. Amount of heat energy required to change the temperature of any substance is given by Q = mcΔt where, m = mass of the substance, c = specific heat of the substance and Δt = change in temperature.
Thermal (Heat) Capacity Heat capacity of any body is equal to the amount of heat energy required to increase its temperature through 1°C. Heat capacity = me where c = specific heat of the substance of the body and m = mass of the body. Its SI unit is joule/kelvin (J/K). Water Equivalent It is the quantity of water whose thermal capacity is same as the heat capacity of the body. It is denoted by W. W = ms = heat capacity of the body. Latent Heat p V The heat energy absorbed or released at constant temperature per unit mass for change of state is called latent heat. Heat energy absorbed or released during change of state is given by Q = mL where m = mass of the substance and L = latent heat. Its unit is cal/g or J/kg and its dimension is [L T ]. For water at its normal boiling point or condensation temperature (100°C), the latent heat of vaporisation is L = 540 cal/g = 40.8 kJ/ mol = 2260 kJ/kg For water at its normal freezing temperature or melting point (0°C), the latent heat of fusion is L = 80 cal/ g = 60 kJ/mol = 336 kJ/kg It is more painful to get burnt by steam rather than by boiling was 100°C gets converted to water at 100°C, then it gives out 536 heat. So, it is clear that steam at 100°C has more heat than wat 100°C (i.e., boiling of water). After snow falls, the temperature of the atmosphere becomes very This is because the snow absorbs the heat from the atmosphere to down. So, in the mountains, when snow falls, one does not feel too but when ice melts, he feels too cold. There is more shivering effect of ice cream on teeth as compare that of water (obtained from ice). This is because when ice cream down, it absorbs large amount of heat from teeth. Melting Conversion of solid into liquid state at constant temperature is melting. Evaporation 2 -2 Conversion of liquid into vapour at all temperatures (even below boiling point) is called evaporation. Boiling
When a liquid is heated gradually, at a particular temperature saturated vapour pressure of the liquid becomes equal to atmospheric pressure, now bubbles of vapour rise to the surface d liquid. This process is called boiling of the liquid. The temperature at which a liquid boils, is called boiling point The boiling point of water increases with increase in pre sure decreases with decrease in pressure. Sublimation The conversion of a solid into vapour state is called sublimation. Hoar Frost The conversion of vapours into solid state is called hoar fr.. Calorimetry This is the branch of heat transfer that deals with the measorette heat. The heat is usually measured in calories or kilo calories. Principle of Calorimetry When a hot body is mixed with a cold body, then heat lost by ha is equal to the heat gained by cold body. Heat lost = Heat gain
Thermal Expansion Increase in size on heating is called thermal expansion. There are three types of thermal expansion.
1. Expansion of solids
2. Expansion of liquids
3. Expansion of gases Expansion of Solids
Three types of expansion -takes place in solid.
Linear Expansion Expansion in length on heating is called linear expansion. Increase in length l = l (1 + α Δt) where, l and l are initial and final lengths,Δt = change in temperature and α = coefficient of linear expansion. Coefficient of linear expansion α = (Δl/l * Δt) where 1= real length and Δl = change in length and Δt= change in temperature. Superficial Expansion Expansion in area on heating is called superficial expansion. Increase in area A = A (1 + β Δt) where, A and A are initial and final areas and β is a coefficient of superficial expansion. Coefficient of superficial expansion β = (ΔA/A * Δt) where. A = area, AA = change in area and At = change in temperature. Cubical Expansion Expansion in volume on heating is called cubical expansion. Increase in volume V = V (1 + γΔt) where V and V are initial and final volumes and γ is a coefficient of cubical expansion.
Coefficient of cubical expansion where V = real volume, AV =change in volume and Δt = change in temperature. Relation between coefficients of linear, superficial and cubical expansions β = 2α and γ = 3α Or α:β:γ = 1:2:3 2.
Expansion of Liquids In liquids only expansion in volume takes place on heating.
(i) Apparent Expansion of Liquids When expansion of th container containing liquid, on heating is not taken into accoun then observed expansion is called apparent expansion of liquids. Coefficient of apparent expansion of a liquid
(ii) Real Expansion of Liquids When expansion of the container, containing liquid, on heating is also taken into account, then observed expansion is called real expansion of liquids. Coefficient of real expansion of a liquid Both, y , and y are measured in °C . We can show that y = y + y where, y , and y are coefficient of real and apparent expansion of liquids and y is coefficient of cubical expansion of the container. Anamalous Expansion of Water When temperature of water is increased from 0°C, then its vol decreases upto 4°C, becomes minimum at r 4°C and then increases. behaviour of water around 4°C is called, anamalous expansion water.
3. Expansion of Gases There are two types of coefficient of expansion in gases (i) Volume Coefficient (γv) At constant pressure, the change in volume per unit volume per degree celsius is called volume coefficient. where V , V , and V are volumes of the gas at 0°C, t °C and t °C. (ii) Pressure Coefficient (γ ) At constant volume, the change in pressure per unit pressure per degree celsius is called pressure coefficient. where p , p and p are pressure of the gas at 0°C, t ° C and t ° C. Practical Applications of Expansion 1. When rails are laid down on the ground, space is left between the end of two rails. 2. The transmission cables are not tightly fixed to the poles. 3. The iron rim to be put on a cart wheel is always of slightly smaller diameter than that of wheel. 4. A glass stopper jammed in the neck of a glass bottle can be taken out by warming the neck of the bottles. Important Points Due to increment in its time period a pendulum clock becomes slow in summer and will lose time. Loss of time in a time period ΔT =(1/2)α ΔθT ∴ Loss of time in any given time interval t can be given by ΔT =(1/2)α Δθt At some higher temperature a scale will expand and scale reading will be lesser than true values, so that true value = scale reading (1 + α Δt) Here, Δt is the temperature difference. However, at lower temperature scale reading will be more or true value will be less
The branch of physics which deals with the study of transformation of heat energy into other forms of energy and vice-versa.
A thermodynamical system is said to be in thermal equilibrium when macroscopic variables (like pressure, volume, temperature, mass, composition etc) that characterise the system do not change with time.
Thermodynamical System An assembly of an extremely large number of particles whose state can be expressed in terms of pressure, volume and temperature, is called thermodynamic system.
Thermodynamic system is classified into the following three systems
(i) Open System It exchange both energy and matter with surrounding.
(ii) Closed System It exchanges only energy (not matter) with surroundings.
(iii) Isolated System It exchanges neither energy nor matter with the surrounding.
A thermodynamic system is not always in equilibrium. For example, a gas allowed to expand freely against vacuum. Similary, a mixture of petrol vapour and air, when ignited by a spark is not an equilibrium state. Equilibrium is acquired eventually with time. Thermodynamic Parameters or Coordinates or Variables The state of thermodynamic system can be described by specifying pressure, volume, temperature, internal energy and number of moles, etc. These are called thermodynamic parameters or coordinates or variables.
Work done by a thermodynamic system is given by W = p * ΔV where p = pressure and ΔV = change in volume. Work done by a thermodynamic system is equal to the area enclosed between the p-V curve and the volume axis Work done in process A-B = area ABCDA Work done by a thermodynamic system depends not only upon the initial and final states of the system but also depend upon the path followed in the process. Work done by the Thermodynamic System is taken as Positive → 4 as volume increases. Negative → 4 as volume decreases. Internal Energy (U) The total energy possessed by any system due to molecular motion and molecular configuration, is called its internal energy. Internal energy of a thermodynamic system depends on temperature. It is the characteristic property of the state of the system.
Zeroth Law of Thermodynamics According to this law, two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other. Thus, if A and B are separately in equilibrium with C, that is if T = T and T = T , then this implies that T = T i.e., the systems A and B are also in thermal equilibrium.
First Law of Thermodynamics Heat given to a thermodynamic system (ΔQ) is partially utilized in doing work (ΔW) against the surrounding and the remaining part increases the internal energy (ΔU) of the system. Therefore, ΔQ = ΔU + ΔW A C B C A B First law of thermodynamics is a restatement of the principle conservation of energy. In isothermal process, change in internal energy is zero (ΔU = 0). Therefore, ΔQ = ΔW In adiabatic process, no exchange of heat takes place, i.e., Δθ = O. Therefore, ΔU = – ΔW In adiabatic process, if gas expands, its internal energy and hence, temperature decreases and vice-versa. In isochoric process, work done is zero, i.e., ΔW = 0, therefore ΔQ = ΔU
Thermodynamic Processes A thermodynamical process is said to take place when some changes’ occur in the state of a thermodynamic system i.e., the therrnodynamie parameters of the system change with time.
(i) Isothermal Process A process taking place in a thermodynamic system at constant temperature is called an isothermal process. Isothermal processes are very slow processes. These process follows Boyle’s law, according to which pV = constant From dU = nC dT as dT = 0 so dU = 0, i.e., internal energy is constant. From first law of thermodynamic dQ = dW, i.e., heat given to the system is equal to the work done by system surroundings. Work done W = 2.3026μRT l0g (V / V ) = 2.3026μRT l0g (p / p ) where, μ = number of moles, R = ideal gas constant, T = absolute temperature and V V and P , P are initial volumes and pressures. After differentiating P V = constant, we have i.e., bulk modulus of gas in isothermal process, β = p. P – V curve for this persons is a rectangular hyperbola Examples (a) Melting process is an isothermal change, because temperature of a substance remains constant during melting. (b) Boiling process is also an isothermal operation.
(ii) Adiabatic Process A process taking place in a thermodynamic system for which there is no exchange of heat between the system and its surroundings. Adiabatic processes are very fast processes. These process follows Poisson’s law, according to which From dQ = nCdT, C = 0 as dQ = 0, i.e., molar heat capacity for adiabatic process is zero. From first law, dU = – dW, i.e., work done by the system is equal to decrease in internal energy. When a system expands adiabatically, work done is positive and hence internal energy decrease, i.e., the system cools down and vice-versa. Work done in an adiabatic process is adi where T and T are initial and final temperatures. Examples (a) Sudden compression or expansion of a gas in a container with perfectly non-conducting wall. (b) Sudden bursting of the tube of a bicycle tyre. (c) Propagation of sound waves in air and other gases.
(iii) Isobaric Process A process taking place in a thermodynamic system at constant pressure is called an isobaric process. Molar heat capacity of the process is C and dQ = nC dT. Internal energy dU = nC dT From the first law of thermodynamics dQ = dU + dW dW = pdV = nRdT Process equation is V / T = constant. p- V curve is a straight line parallel to volume axis.
(iv) Isochoric Process A process taking place in a tlaermodynars system at constant volume is called an isochoric process. dQ = nC dT, molar heat capacity for isochoric process is C . Volume is constant, so dW = 0, Process equation is p / T = constant p- V curve is a straight line parallel to pressure axis. i f p p v v v (v) Cyclic Process When a thermodynamic system returns to . initial state after passing through several states, then it is called cyclic process.
Efficiency of the cycle is given by Work done by the cycle can be computed from area enclosed cycle on p- V curve. Isothermal and Adiabatic Curves The graph drawn between the pressure p and the volume V of a given mass of a gas for an isothermal process is called isothermal curve and for an adiabatic process it is called adiabatic curve . The slope of the adiabatic curve = γ x the slope of the isothermal curve Volume Elasticities of Gases
There are two types of volume elasticities of gases (i) Isothermal modulus of elasticity E = p (ii) Adiabatic modulus of elasticity E = γ p Ratio between isothermal and adiabatic modulus E / E = γ = C / C where C and C are specific heats of gas at constant pressure and at constant volume. For an isothermal process Δt = 0, therefore specific heat, c = Δ θ / m Δt = &infi; For an adiabatic process 119= 0, therefore specific heat, c = 0 / m Δt = 0
Second Law of Thermodynamics The second law of thermodynamics gives a fundamental limitation to the efficiency of a heat engine and the coefficient of performance of a refrigerator. It says that efficiency of a heat engine can never be unity (or 100%). This implies that heat released to the cold reservoir can never be made zero.
Kelvin’s Statement It is impossible to obtain a continuous supply of work from a body by cooling it to a temperature below the coldest of its surroundings. Clausius’ Statement It is impossible to transfer heat from a lower temperature body to a higher temperature body without use of an extemal agency. Planck’s Statement It is impossible to construct a heat engine that will convert heat completely into work. All these statements are equivalent as one can be obtained from the other.
Entropy Entropy is a physical quantity that remains constant during a reversible adiabatic change. Change in entropy is given by dS = δQ / T p v Where, δQ = heat supplied to the system and T = absolute temperature. Entropy of a system never decreases, i.e., dS ≥ o. Entropy of a system increases in an irreversible process
Heat Engine A heat energy engine is a device which converts heat energy into mechanical energy. A heat engine consists of three part
(i) Source of heat at higher temperature
(ii) Working substance
(iii) Sink of heat at lower temperature Thermal efficiency of a heat engine is given by where Q is heat absorbed from the source, Q is heat rejected to the sink and T and T are temperatures of source and sink.
Heat engine are of two types
(i) External Combustion Engine In this engine fuel is burnt a chamber outside the main body of the engine. e.g., steam engine. In practical life thermal efficiency of a steam engine varies from 12% to 16%.
(ii) Internal Combustion Engine In this engine. fuel is burnt inside the main body of the engine. e.g., petrol and diesel engine. In practical life thermal efficiency of a petrol engine is 26% and a diesel engine is 40%. Carnot’s Cycle Carnot devised an ideal cycle of operation for a heat engine, called Carnot’s cycle. A Carnot’s cycle contains the following four processes (i) Isothermal expansion (AB) (ii) Adiabatic expansion (BO) (iii) Isothermal compression (CD) (iv) Adiabatic compression (DA) The net work done per cycle by the engine is numerically equal to the area of the loop representing the Carnot’s cycle . After doing the calculations for different processes we can show that [Efficiency of Carnot engine is maximum (not 1000/0) for given temperatures T and T . But still Carnot engine is not a practical 1 2 engine because many ideal situations have been assumed while designing this engine which can practically not be obtained.]
Refrigerator or Heat Pump A refrigerator or heat pump is a device used for cooling things. It absorb heat from sink at lower temperature and reject a larger amount of heat to source at higher temperature. Coefficient of performance of refrigerator is given by where Q is heat absorbed from the sink, Q is heat rejected to source and T and T are temperatures of source and sink