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Systems of Particles and Rotational Motion Class 11 Notes Physics Chapter 7
• A rigid body is a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of such a body do not change.
• Centre of Mass
For a system of particles, the centre of mass is defined as that point where the entire mass of the system is imagined to be concentrated, for consideration of its translational motion.
If all the external forces acting on the body/system of bodies were to be applied at the centre of mass, the state of rest/ motion of the body/system of bodies shall remain unaffected.
• The centre of mass of a body or a system is its balancing point. The centre of mass of a two- particle system always lies on the line joining the two particles and is somewhere in between the particles.
• Motion of centre of Mass
The centre of mass of a system of particles moves as if the entire mass of the system were concentrated at the centre of mass and all the external forces were applied at that point. Velocity of centre of mass of a system of two particles, m1 and m2 with velocity v1 and v2 is given
• If no external force acts on the body, then the centre of mass will have constant momentum. Its velocity is constant and acceleration is zero, i.e., MVcm = constant.
• Vector Product or Cross Product of two vectors
Torque is the moment of force. Torque acting on a particle is defined as the product of the magnitude of the force acting on the particle and the perpendicular distance of the application of force from the axis of rotation of the particle.
• Angular Momentum
The angular momentum (or moment of momentum) about an axis of rotation is a vector quantity, whose magnitude is equal to the product of the magnitude of momentum and the perpendicular distance of the line of action of momentum from the axis of rotation and its direction is perpendicular to the plane containing the momentum and the perpendicular distance.
• Axis of Rotation
A rigid body is said to be rotating if every point mass that makes it up, describes a circular path of a different radius but the same angular speed. The circular paths of all the point masses have a common centre. A line passing through this common centre is the axis of rotation.
• A rigid body is said to be in equilibrium if under the action of forces/torques, the body remains in its position of rest or of uniform motion.
For translational equilibrium, the vector sum of all the forces acting on a body must be zero. For rotational equilibrium, the vector sum of torques of all the forces acting on that body about the reference point must be zero. For complete equilibrium, both these conditions must be fulfilled.
Two equal and opposite forces acting on a body but having different lines of action, form a couple. The net force due to a couple is zero, but they exert a torque and produce rotational motion.
• Moment of Inertia
The rotational inertia of a rigid body is referred to as its moment of inertia.
The moment of inertia of a body about an axis is defined as the sum of the products of the masses of the particles constituting the body and the square of their respective perpendicular distance from the axis.
It is given by .
• Radius of Gyration
The distance of a point in a body from the axis of rotation, at which if whole of the mass of the body were supposed to be concentrated, its moment of inertia about the axis of rotation would be the same as that determined by the actual distribution of mass of the body is called radius of gyration.
If we consider that the whole mass of the body is concentrated at a distance K from the axis of rotation, then moment of inertia I can be expressed as I = MK2
• Theorem of Parallel Axes
According to this theorem, the moment of inertia I of a body about any axis is equal to its moment of inertia about a parallel axis through centre of mass, Icm, plus Ma2 where M is the mass of the body and V is the perpendicular distance between the axes, i.e.,
I = Icm + Ma2
• Theorem of Perpendicular Axes
According to this theorem, the moment of inertia I of the body about a perpendicular axis is equal to the sum of moments of inertia of the body about two axes at right angles to each other in the plane of the body and intersecting at a point where the perpendicular axis passes, i.e.,
• Rolling Motion
The combination of rotational motion and the translational motion of a rigid body is known as rolling motion.
• Law of Conservation of Angular Momentum
According to the law of conservation of angular momentum, if there is no external couple acting, the total angular momentum of a rigid body or a system of particles is conserved.
• IMPORTANT TABLES
CBSE Class 11 Physics Chapter-7 Important Questions
1 Marks Questions
1.A wheel 0.5m in radius is moving with a speed of 12m/s. find its angular speed?
2.State the condition for translational equilibrium of a body?
Ans:For translations equilibrium of a body the vector sum of all the forces acting on the body must be zero.
3.How is angular momentum related to linear momentum?
Where is the angle between
4.What is the position of the centre of mass of a uniform triangular lamina?
Ans:At the centroid of the triangular lamina.
5.What is the moment of inertia of a sphere of mass 20and radius about its diameter?
I = 0.5kgm2
6.What are the factors on which moment of inertia of a body depends?
Ans: (1) Mass of the body
(2) Shape and size of the body
(3) Position of the axis of rotation
7.Two particles in an isolated system under go head on collision. What is the acceleration of the centre of mass of the system?
Ans: Acceleration is zero as force, are internal forces.
8.Which component of a force does not contribute towards torque?
Ans: The radial component of a force does not contribute towards torque.
9.What is the position of centre of mass of a rectangular lamina?
Ans: The centre of mass of a rectangular lamina is the point of intersection of diagonals.
10. Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?
Ans.Geometric centre; No
The centre of mass (C.M.) is a point where the mass of a body is supposed to be concentrated. For the given geometric shapes having a uniform mass density, the C.M. lies at their respective geometric centers.
The centre of mass of a body need not necessarily lie within it. For example, the C.M. of bodies such as a ring, a hollow sphere, etc., lies outside the body.
11. A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system?
Ans. No change
The child is running arbitrarily on a trolley moving with velocity v. However, the running of the child will produce no effect on the velocity of the centre of mass of the trolley. This is because the force due to the boy’s motion is purely internal. Internal forces produce no effect on the motion of the bodies on which they act. Since no external force is involved in the boy-trolley system, the boy’s motion will produce no change in the velocity of the centre of mass of the trolley.
12. To maintain a rotor at a uniform angular speed of 200 rad, an engine needs to transmit a torque of 180 Nm. What is the power required by the engine?
(Note: uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is 100 % efficient.
Ans. Angular speed of the rotor, = 200 rad/s
Torque required, = 180 Nm
The power of the rotor (P) is related to torque and angular speed by the relation:
= 180 200 = 36 103
= 36 kW
Hence, the power required by the engine is 36 kW.
1 MARK QUESTIONS
- What Is a rigid body?
- State the principle of moments of rotational equilibrium.
- Is centre of mass of a body necessarily lie inside the body? Give any example
Ans. No. example ring
- Can the couple acting on a rigid body produce translatory motion?
Ans. No. It can produce only rototory motion.
- Which component of linear momentum does not contribute to angular momentum?
Ans. Radial Component
- A system is in stable equilibrium. What can we say about Its potential energy?
Ans. P.E is minimum.
- Is radius of gyration a constant quantity?
Ans. No, it changes with the position of axis of rotation.
- Two solid spheres of the same mass are made of metals of different densities. Which of them has a large moment of inertia about the diameter?
Ans. Sphere of small density will have large moment of inertia.
- The moment of inertia of two rotating bodies A and B are lA and lB (lA > lB) and their angular momenta are equal. Which one has a greater kinetic energy?
- A particle moves on a circular path with decreasing speed. What happens to its angular momentum?
Ans. as = i:e magnitude decreases but direction remains constant.
- What is the value of instantaneous speed of the point of contact during pure rolling?
- Which physical quantity is conserved when a planet revolves around the sun?
Ans. Angular momentum of planet.
- What is the value of torque on the planet due to the gravitational force of sun?
- If no external torque acts on a body, will its angular velocity be constant?
Ans. No. wot Y.
- Why there are two propellers in a helicopter?
Ans. due to conservation of angular momentum
- A child sits stationary at one end of a long trolley moving uniformly with speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, then what is the effect of the speed of the centre of mass of the (trolley -t- child) system?
Ans. No change in speed of system as no external force is working.
2 Marks Questions
1.A planet revolves around on massive star in a highly elliptical orbit is its angular momentum constant over the entire orbit. Give reason?
Ans: A planet revolves around the star under the effect of gravitational force since the force is radial and does not contribute towards torque. Thus in the absence of an external torque angular momentum of the planet remains constant.
2.Obtain the equation?
Integrating within the limits
3.What is the torque of the force acting at the point about the origin?
4.What is the value of linear velocity if
5.Establish the third equation of rotational motion
Multiply and divide by
Integrating we get
6.Find the expression for radius of gyration of a solid sphere about one of its diameter?
Ans:M.I of a solid sphere
About its diameter =
K = Radius of Gyration
7.Prove that the centre of mass of two particles divides the line joining the particles in the inverse ratio of their masses?
If centre of mass is at the origin
In terms of magnitude
8.Show that cross product of two parallel vectors is zero?
If and are parallel to each other
9.Prove the relation
Differentiating wrt. Time
From (1) and (2)
10.Show that for an isolated system the centre of mass moves with uniform velocity along a straight line path?
Ans: Let be the total mass concentrated at centre of mass whose position vector is
For an isolated system
11.The angle θ covered by a body in rotational motion is give by the equation θ = 6t + 5t2 + 2t3. Determine the value of instantaneous angular velocity and angular acceleration at time t = 2S.
Again angular acceleration
12. A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad . The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?
Ans.Mass of the cylinder, m= 20 kg
Angular speed, = 100 rad
Radius of the cylinder, r= 0.25 m
The moment of inertia of the solid cylinder:
∴Angular momentum, L= I
= 6.25 100
= 62.5 Js
13. A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping.
Ans.Mass of the hollow cylinder, m= 3 kg
Radius of the hollow cylinder, r= 40 cm = 0.4 m
Applied force, F = 30 N
The moment of inertia of the hollow cylinder about its geometric axis:
= 30 0.4 = 12 Nm
For angular acceleration, torque is also given by the relation:
Linear acceleration = rα= 0.4 25 = 10 m
14. A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it.
(Hint: The moment of inertia of the door about the vertical axis at one end is.)
Ans. Mass of the bullet, m= 10 g =
Velocity of the bullet, v= 500 m/s
Thickness of the door, L= 1 m
Radius of the door,
Mass of the door, M= 12 kg
Angular momentum imparted by the bullet on the door:
Moment of inertia of the door:
15. Explain why friction is necessary to make the disc in Fig. 7.41 roll in the direction indicated.
(a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.
(b) What is the force of friction after perfect rolling begins?
Ans. A torque is required to roll the given disc. As per the definition of torque, the rotating force should be tangential to the disc. Since the frictional force at point B is along the tangential force at point A, a frictional force is required for making the disc roll.
(a) Force of friction acts opposite to the direction of velocity at point B. The direction of linear velocity at point B is tangentially leftward. Hence, frictional force will act tangentially rightward. The sense of frictional torque before the start of perfect rolling is perpendicular to the plane of the disc in the outward direction.
(b) Since frictional force acts opposite to the direction of velocity at point B, perfect rolling will begin when the velocity at that point becomes equal to zero. This will make the frictional force acting on the disc zero.
2 MARKS QUESTIONS
- Show that in the absence of any external force, the velocity of the centre of mass remains constant.
- State the factors on which the position of centre of mass of a rigid body depends.
(i) Shape of body
(ii) mass distribution
- What is the turning effect of force called for? On what factors does it depend?
(i) Magnitude of force
(ii) Perpendicular distance of force vector from axis of rotation.
- State the factors on which the moment of inertia of a body depends.
(I) Mass of body
(ii) Size and shape of body
(iii) Mass distribution w.r.t. axis of rotation
(iv) position and orientation of rotational axis
- On what factors does radius of gyration of body depend?
Ans. Mass distribution.
- Why do we prefer to use a wrench of longer arm?
Ans. to increase torque.
- Can a body be In equilibrium while in motion? If yes, give an example
Ans. Yes, if body has no linear and angular acceleration. Hence a body in uniform straight line motion will be in equilibrium.
- There is a stick half of which is wooden and half is of steel,
- it is pivoted at the wooden end and a force is applied at the steel end at right angle to its length
- it is pivoted at the steel end and the same force is applied at the wooden end. In which case is the angular acceleration more and why?
Ans. I (first case) > l(Second case)
(first case) < ( Second case)
- If earth contracts to half its radius what would be the length of the day at equator?
- An internal force cannot change the state of motion of centre of mass of a body. How does the internal force of the brakes bring a vehicle to rest?
Ans. In this case the force which bring the vehicle to rest is friction, and it is an external force.
- When does a rigid body said to be in equilibrium? State the necessary condition for a body to be in equilibrium.
Ans. For translation equilibrium
For rotational equilibrium
- How will you distinguish between a hardboiled egg and a raw egg by spinning it on a table top?
Ans. For same external torque, angular acceleration of raw egg will be small than that of Hardboiled egg
- What are binary stars? Discuss their motion in respect of their centre of mass.
- In which condition a body in gravitational field Is in stable equilibrium?
Ans. When vertical line through centre of gravity passes through the base of the body.
- Give the physical significance of moment of inertia.
Ans. It plays the same role in rotatory motion as the mass does in translatory motion.
3 Marks Questions
1.The moment of inertia of a solid sphere about a tangent is. Find the moment of inertia about a diameter?
Ans: A tangent KCl is drawn at pt. C of a solid sphere of mass M and radius R. Draw a diameter AOB || to KCl.
Then according to Theorem of parallel axis, I, = I + M (OC)2
I1(M.I about the tangent) =
2.Four particles of mass 1kg, 2kg, 3kg and 4kg are placed at the four vertices A, B, C and D of square of side 1m. Find the position of centre of mass of the particle.
Thus centre of mass (0.5m, 0.7m)
3.A circular ring of diameter 40cm and mass 1kg is rotating about an axis normal to its plane and passing through the centre with a frequency of 10 rotations per second. Calculate the angular momentum about its axis of rotation?
4.(a) Which physical quantities are represented by the
(i) Rate of change of angular momentum
(ii) Product of I and
(b) Show that angular momentum of a satellite of mass MS revolving around the earth having mass Me in an orbit of radius r is equal to
Ans:(a) (1) Torque i.e.
(2) Angular momentum i.e. L = Iw
(b) Mass of satellite = Ms
Mass of earth = Me
Radius of satellite = r
Required centripetal force
Where is the orbital velocity with which the satellite revolves round the earth.
Gravitational force between the satellite and the earth
Equating (1) and (2)
Now angular momentum of the satellite
5. In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 . Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.
Ans.The given situation can be shown as:
Distance between H and Cl atoms = 1.27
Mass of H atom = m
Mass of Cl atom = 35.5m
Let the centre of mass of the system lie at a distance x from the Cl atom.
Distance of the centre of mass from the H atom = (1.27 – x)
Let us assume that the centre of mass of the given molecule lies at the origin. Therefore, we can have:
Here, the negative sign indicates that the centre of mass lies at the left of the molecule.
Hence, the centre of mass of the HCl molecule lies 0.037 from the Cl atom.
6. Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a x b.
Ans.Consider two vectors and, inclined at an angle θ, as shown in the following figure.
In ΔOMN, we can write the relation:
= 2 × Area of ΔOMK
∴Area of ΔOMK
7. A metre stick is balanced on a knife edge at its centre. When two coins, each of mass 5 g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm. What is the mass of the metre stick?
Ans. Let W and be the respective weights of the metre stick and the coin.
The mass of the metre stick is concentrated at its mid-point, i.e., at the 50 cm mark.
Mass of the meter stick =
Mass of each coin, m= 5 g
When the coins are placed 12 cm away from the end P, the centre of mass gets shifted by 5 cm from point R toward the end P. The centre of mass is located at a distance of 45 cm from point P.
The net torque will be conserved for rotational equilibrium about point R.
Hence, the mass of the metre stick is 66 g.
4 Marks Questions
1. Show that a.(b c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a, b and c.
Ans. A parallelepiped with origin O and sides a, b, and c is shown in the following figure.
Volume of the given parallelepiped = abc
Let be a unit vector perpendicular to both band c. Hence, and a have the same direction.
= abc cosθ
= abc cos 0°
= Volume of the parallelepiped.
2. A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?
Ans. Radius of the hoop, r= 2 m
Mass of the hoop, m= 100 kg
Velocity of the hoop, v= 20 cm/s = 0.2 m/s
Total energy of the hoop = Translational KE + Rotational KE
Moment of inertia of the hoop about its centre, I =
But we have the relation,
The work required to be done for stopping the hoop is equal to the total energy of the hoop.
∴Required work to be done,
3. The oxygen molecule has a mass of kg and a moment of inertia of about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500 m/s and that its kinetic energy of rotation is two thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.
Ans. Mass of an oxygen molecule, m= kg
Moment of inertia, I=
Velocity of the oxygen molecule, v= 500 m/s
The separation between the two atoms of the oxygen molecule = 2r
Mass of each oxygen atom =
Hence, moment of inertia I, is calculated as:
It is given that:
4. A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg.
(a) What is his new angular speed? (Neglect friction.)
(b) Is kinetic energy conserved in the process? If not, from where does the change come about?
Ans. (a) 58.88 rev/min (b) No
(a)Moment of inertia of the man-platform system = 7.6 kg
Moment of inertia when the man stretches his hands to a distance of 90 cm:
= 8.1 kg
Initial moment of inertia of the system,
Angular momentum, …………(i)
Moment of inertia when the man folds his hands to a distance of 20 cm:
= = 0.4 kg
Final moment of inertia,
Final angular speed =
Final angular momentum, … (ii)
From the conservation of angular momentum, we have:
(b) Kinetic energy is not conserved in the given process. In fact, with the decrease in the moment of inertia, kinetic energy increases. The additional kinetic energy comes from the work done by the man to fold his hands toward himself.
5. Read each statement below carefully, and state, with reasons, if it is true or false;
(a) During rolling, the force of friction acts in the same direction as the direction of motion of the CM of the body.
(b) The instantaneous speed of the point of contact during rolling is zero.
(c) The instantaneous acceleration of the point of contact during rolling is zero.
(d) For perfect rolling motion, work done against friction is zero.
(e) A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion.
Ans. (a) False
Frictional force acts opposite to the direction of motion of the centre of mass of a body. In the case of rolling, the direction of motion of the centre of mass is backward. Hence, frictional force acts in the forward direction.
Rolling can be considered as the rotation of a body about an axis passing through the point of contact of the body with the ground. Hence, its instantaneous speed is zero.
When a body is rolling, its instantaneous acceleration is not equal to zero. It has some value.
When perfect rolling begins, the frictional force acting at the lowermost point becomes zero. Hence, the work done against friction is also zero.
The rolling of a body occurs when a frictional force acts between the body and the surface. This frictional force provides the torque necessary for rolling. In the absence of a frictional force, the body slips from the inclined plane under the effect of its own weight.
6. Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the vector angular momentum of the two particle system is the same whatever be the point about which the angular momentum is taken.
Ans. Let at a certain instant two particles be at points P and Q, as shown in the following figure.
Angular momentum of the system about point P:
Angular momentum of the system about point:
Consider a point R, which is at a distance y from point Q, i.e.,
QR = y
∴PR = d – y
Angular momentum of the system about point R:
Comparing equations (i), (ii), and (iii), we get:
We infer from equation (iv) that the angular momentum of a system does not depend on the point about which it is taken.
5 Marks Questions
1.(a) Why is moment of inertia called rotational inertia?
(b) Calculate M.I of a uniform circular disc of mass 500gm and radius 10cm about
(i) Diameter (ii) axis tangent to the disc and parallel to diameter
(c) Axis passing trough centre and perpendicular to its plane?
Ans: (a) Moment of inertia is called rotational inertia because it measures moment of inertia during its rotational motion.
2.(a) A cat is able to land on its feet after a fall. Why?
(b) If angular momentum moment of inertia is decreased, will its rotational be also conserved? Explain.
Ans:(a) When cat lands to the ground, if stretches its tail as result M.I increases
As IW = constant
Angular speed will be small due to increase in moment of inertia and the cat is able to land on its feet without any harm.
(b) Let moment of inertia of a system decrease from I to I’
Then angular speed increase from w to w’
K.E. of rotation of the system
K.E of the system will increase. Hence it will not be conserved.
3. Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components. Show that if the particle moves only in the x–y plane the angular momentum has only a z-component.
Linear momentum of the particle,
Position vector of the particle,
Comparing the coefficients of we get:
The particle moves in the x–y plane. Hence, the z-component of the position vector and linear momentum vector becomes zero, i.e.,
z= = 0
Thus, equation (i) reduces to:
Therefore, when the particle is confined to move in the x–y plane, the direction of angular momentum is along the z-direction.
4. A non-uniform bar of weight W is suspended at rest by two strings of negligible weight as shown in Fig.7.39. The angles made by the strings with the vertical are 36.9° and 53.1° respectively. The bar is 2 m long. Calculate the distance d of the centre of gravity of the bar from its left end.
Ans. The free body diagram of the bar is shown in the following figure.
Length of the bar, l= 2 m
and are the tensions produced in the left and right strings respectively.
At translational equilibrium, we have:
For rotational equilibrium, on taking the torque about the centre of gravity, we have:
Hence, the C.G. (centre of gravity) of the given bar lies 0.72 m from its left end.
5.A car weighs 1800 kg. The distance between its front and back axles is 1.8 m. Its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.
Ans. Mass of the car, m= 1800 kg
Distance between the front and back axles, d= 1.8 m
Distance between the C.G. (centre of gravity) and the back axle = 1.05 m
The various forces acting on the car are shown in the following figure.
and are the forces exerted by the level ground on the front and back wheels respectively.
At translational equilibrium:
= 1800 9.8
= 17640 N … (i)
For rotational equilibrium, on taking the torque about the C.G., we have:
Solving equations (i) and (ii), we get:
∴ = 17640 – 7350 = 10290 N
Therefore, the force exerted on each front wheel, and
The force exerted on each back wheel
6. (a) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be, where M is the mass of the sphere and R is the radius of the sphere.
(b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.
The moment of inertia (M.I.) of a sphere about its diameter =
According to the theorem of parallel axes, the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.
The M.I. about a tangent of the sphere =
The moment of inertia of a disc about its diameter =
According to the theorem of perpendicular axis, the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.
The M.I. of the disc about its centre =
The situation is shown in the given figure.
Applying the theorem of parallel axes:
The moment of inertia about an axis normal to the disc and passing through a point on its edge =
7. Torques of equal magnitude is applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?
Ans. Let m and r be the respective masses of the hollow cylinder and the solid sphere.
The moment of inertia of the hollow cylinder about its standard axis,
The moment of inertia of the solid sphere about an axis passing through its centre,
We have the relation:
α = Angular acceleration
T = Torque
I = Moment of inertia
For the hollow cylinder,
For the solid sphere,
As an equal torque is applied to both the bodies,
Now, using the relation:
= Initial angular velocity
t = Time of rotation
= Final angular velocity
For equal and t, we have:
From equations (i) and (ii), we can write:
Hence, the angular velocity of the solid sphere will be greater than that of the hollow cylinder.
8. (a) A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of 40 rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/5 times the initial value? Assume that the turntable rotates without friction.
(b) Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?
Initial angular velocity, = 40 rev/min
Final angular velocity =
The moment of inertia of the boy with stretched hands =
The moment of inertia of the boy with folded hands =
The two moments of inertia are related as:
Since no external force acts on the boy, the angular momentum L is a constant.
Hence, for the two situations, we can write:
(b)Final K.E. = 2.5 Initial K.E.
Final kinetic rotation,
Initial kinetic rotation,
The increase in the rotational kinetic energy is attributed to the internal energy of the boy.
9. From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.
Ans. R/6; from the original centre of the body and opposite to the centre of the cut portion.
Mass per unit area of the original disc =
Radius of the original disc = R
Mass of the original disc, M =
The disc with the cut portion is shown in the following figure:
Radius of the smaller disc =
Mass of the smaller disc, M‘ =
Let O and be the respective centers of the original disc and the disc cut off from the original. As per the definition of the centre of mass, the centre of mass of the original disc is supposed to be concentrated at O, while that of the smaller disc is supposed to be concentrated at .
It is given that:
After the smaller disc has been cut from the original, the remaining portion is considered to be a system of two masses. The two masses are:
M (concentrated at O), and
(The negative sign indicates that this portion has been removed from the original disc.)
Let x be the distance through which the centre of mass of the remaining portion shifts from point O.
The relation between the centers of masses of two masses is given as:
For the given system, we can write:
(The negative sign indicates that the centre of mass gets shifted toward the left of point O.)
10. A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c) If so, which one and why?
Ans. (a) Yes (b)Yes (c) On the smaller inclination
(a)Mass of the sphere = m
Height of the plane = h
Velocity of the sphere at the bottom of the plane =v
At the top of the plane, the total energy of the sphere = Potential energy = mgh
At the bottom of the plane, the sphere has both translational and rotational kinetic energies.
Hence, total energy =
Using the law of conservation of energy, we can write:
For a solid sphere, the moment of inertia about its centre,
Hence, equation (i) becomes:
But we have the relation,
Hence, the velocity of the sphere at the bottom depends only on height (h) and acceleration due to gravity (g). Both these values are constants. Therefore, the velocity at the bottom remains the same from whichever inclined plane the sphere is rolled.
(b), (c) Consider two inclined planes with inclinations and, related as:
The acceleration produced in the sphere when it rolls down the plane inclined at is:
The various forces acting on the sphere are shown in the following figure.
is the normal reaction to the sphere.
Similarly, the acceleration produced in the sphere when it rolls down the plane inclined at is:
The various forces acting on the sphere are shown in the following figure.
is the normal reaction to the sphere.
> ; sin > sin … (i)
∴ > … (ii)
Initial velocity, u = 0
Final velocity, v= Constant
Using the first equation of motion, we can obtain the time of roll as:
v= u + at
11. A solid cylinder rolls up an inclined plane of angle of inclination 30°. At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s.
(a) How far will the cylinder go up the plane?
(b) How long will it take to return to the bottom?
Ans.A solid cylinder rolling up an inclination is shown in the following figure.
Initial velocity of the solid cylinder, v = 5 m/s
Angle of inclination, θ = 30°
Height reached by the cylinder = h
(a) Energy of the cylinder at point A:
Energy of the cylinder at point B = mgh
Using the law of conservation of energy, we can write:
Moment of inertia of the solid cylinder,
But we have the relation,
Hence, the cylinder will travel 3.82 m up the inclined plane.
(b) For radius of gyration K, the velocity of the cylinder at the instance when it rolls back to the bottom is given by the relation:
For the solid cylinder,
The time taken to return to the bottom is:
Therefore, the total time taken by the cylinder to return to the bottom is (2 0.764) 1.53 s.
12. As shown in Fig.7.40, the two sides of a step ladder BA and CA are 1.6 m long and hinged at A. A rope DE, 0.5 m is tied half way up. A weight 40 kg is suspended from a point F, 1.2 m from B along the ladder BA. Assuming the floor to be frictionless and neglecting the weight of the ladder, find the tension in the rope and forces exerted by the floor on the ladder. (Take g = 9.8)
(Hint: Consider the equilibrium of each side of the ladder separately.)
Ans. The given situation can be shown as:
NB = Force exerted on the ladder by the floor point B
NC = Force exerted on the ladder by the floor point C
T = Tension in the rope
BA = CA = 1.6 m
DE = 0. 5 m
BF = 1.2 m
Mass of the weight, m = 40 kg
Draw a perpendicular from A on the floor BC. This intersects DE at mid-point H.
ΔABI and ΔAIC are similar
∴BI = IC
Hence, I is the mid-point of BC.
DE || BC
BC = 2 DE = 1 m
AF = BA – BF = 0.4 m … (i)
D is the mid-point of AB.
Hence, we can write:
Using equations (i) and (ii), we get:
FE = 0.4 m
Hence, F is the mid-point of AD.
FGDH and F is the mid-point of AD. Hence, G will also be the mid-point of AH.
ΔAFG and ΔADH are similar
For translational equilibrium of the ladder, the upward force should be equal to the downward force.
= mg = 392 … (iii)
For rotational equilibrium of the ladder, the net moment about A is:
Adding equations (iii) and (iv), we get:
For rotational equilibrium of the side AB, consider the moment about A.
13. Two discs of moments of inertia and about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds and are brought into contact face to face with their axes of rotation coincident. (a) What is the angular speed of the two-disc system? (b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take ≠.
Moment of inertia of disc
Angular speed of disc
Angular speed of disc
Angular momentum of disc
Angular momentum of disc
Angular momentum of disc
Total initial angular momentum,
When the two discs are joined together, their moments of inertia get added up.
Moment of inertia of the system of two discs,
Let be the angular speed of the system.
Total final angular momentum,
Using the law of conservation of angular momentum, we have:
(b)Kinetic energy of disc I,
Kinetic energy of disc II,
Total initial kinetic energy,
When the discs are joined, their moments of inertia get added up.
Moment of inertia of the system,
Angular speed of the system =
Final kinetic energy:
All the quantities on RHS are positive.
The loss of KE can be attributed to the frictional force that comes into play when the two discs come in contact with each other.
14. (a) Prove the theorem of perpendicular axes.
(Hint: Square of the distance of a point (x, y) in the x–y plane from an axis through the origin perpendicular to the plane is).
(b) Prove the theorem of parallel axes.
(Hint: If the centre of mass is chosen to be the origin ).
Ans.(a)The theorem of perpendicular axes states that the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.
A physical body with centre O and a point mass m, in the x–y plane at (x, y) is shown in the following figure.
Moment of inertia about x-axis, Ix=
Moment of inertia about y-axis, Iy=
Moment of inertia about z-axis, Iz=
Hence, the theorem is proved.
(b)The theorem of parallel axes states that the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.
Suppose a rigid body is made up of n particles, having masses, at perpendicular distances respectively from the centre of mass O of the rigid body.
The moment of inertia about axis RS passing through the point O:
The perpendicular distance of mass mi, from the axis QP = a+ ri
Hence, the moment of inertia about axis QP:
Now, at the centre of mass, the moment of inertia of all the particles about the axis passing through the centre of mass is zero, that is,
M=Total mass of the rigid body
Hence, the theorem is proved.
15. Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by .
Using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.
Ans. A body rolling on an inclined plane of height h, is shown in the following figure:
m= Mass of the body
R = Radius of the body
K= Radius of gyration of the body
v = Translational velocity of the body
h =Height of the inclined plane
g = Acceleration due to gravity
Total energy at the top of the plane, =mgh
Total energy at the bottom of the plane,
From the law of conservation of energy, we have:
Hence, the given result is proved.
16. A disc rotating about its axis with angular speed is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R. What are the linear velocities of the points A, B and C on the disc shown in Fig. 7.41? Will the disc roll in the direction indicated?
Ans. = R; = R; ; The disc will not roll
Angular speed of the disc =
Radius of the disc = R
Using the relation for linear velocity, v = R
For point A:
= R; in the direction tangential to the right
For point B:
= R; in the direction tangential to the left
For point C:
;in the direction same as that of
The directions of motion of points A, B, and C on the disc are shown in the following figure
Since the disc is placed on a frictionless table, it will not roll. This is because the presence of friction is essential for the rolling of a body.
17. A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to 10 π rad. Which of the two will start to roll earlier? The co-efficient of kinetic friction is = 0.2.
Radii of the ring and the disc, r = 10 cm = 0.1 m
Initial angular speed, =10 π rad
Coefficient of kinetic friction, = 0.2
Initial velocity of both the objects, u = 0
Motion of the two objects is caused by frictional force. As per Newton’s second law of motion, we have frictional force, f = ma
a = Acceleration produced in the objects
m = Mass
∴a = g … (i)
As per the first equation of motion, the final velocity of the objects can be obtained as:
v = u + at
= 0 + gt
= gt … (ii)
The torque applied by the frictional force will act in perpendicularly outward direction and cause reduction in the initial angular speed.
Torque, T= –Iα
α = Angular acceleration
Using the first equation of rotational motion to obtain the final angular speed:
Rolling starts when linear velocity, v = r
Equating equations (ii) and (v), we get:
Since, the disc will start rolling before the ring.
18. A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30°. The coefficient of static friction = 0.25.
(a) How much is the force of friction acting on the cylinder?
(b) What is the work done against friction during rolling?
(c) If the inclination of the plane is increased, at what value of does the cylinder begin to skid, and not roll perfectly?
Ans. Mass of the cylinder, m= 10 kg
Radius of the cylinder, r= 15 cm = 0.15 m
Co-efficient of kinetic friction, = 0.25
Angle of inclination, θ= 30°
Moment of inertia of a solid cylinder about its geometric axis,
The various forces acting on the cylinder are shown in the following figure:
The acceleration of the cylinder is given as:
(a) Using Newton’s second law of motion, we can write net force as:
(b) During rolling, the instantaneous point of contact with the plane comes to rest. Hence, the work done against frictional force is zero.
(c) For rolling without skid, we have the relation:
19. Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
Where pi is the momentum of the I th particle (of mass mi) and is the velocity of the I th particle relative to the centre of mass.
Also, prove using the definition of the centre of mass
(b) Show K =
Where K is the total kinetic energy of the system of particles, is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec.7.14.
Where is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass . Remember rest of the notation is the standard notation used in the chapter. Note and can be said to be angular momenta, respectively, about and of the centre if mass of the system of particles.
Further show that
Where is the sum of all external torques acting on the system about the centre of mass.
(Hint: Use the definition of centre of mass and Newton’s Third Law. Assume the internal forces between any two particles act along the line joining the particles.)
Ans. (a)Take a system of i moving particles.
Mass of the ith particle =
Velocity of the ith particle =
Hence, momentum of the ith particle,
Velocity of the centre of mass = V
The velocity of the ith particle with respect to the centre of mass of the system is given as:
Multiplying throughout equation (1), we get:
= Momentum of the ith particle with respect to the centre of mass of the system
We have the relation:
Taking the summation of momentum of all the particles with respect to the centre of mass of the system, we get:
= Position vector of its particle with respect to the centre of mass
As per the definition of the centre of mass, we have:
(b) We have the relation for velocity of the particle as:
Taking the dot product of equation (2) with itself, we get:
Here, for the centre of mass of the system of particles,
Where, K = = Total kinetic energy of the system of particles
K‘ = = Total kinetic energy of the system of particles with respect to the centre of mass
= Kinetic energy of the translation of the system as a whole
- Three masses 3 kg, 4 kg and 5 kg are located at the corners of an equilateral triangle of side 1m. Locate the centre of mass of the system.
Ans. (x,y)=(0.54 m, 0.36m)
- Two particles mass 100 g and 300 g at a given time have velocities and ms-1 respectively. Determine velocity of COM.
Ans. Velocity of COM=
- From a uniform disc of radius R, a circular disc of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of original disc. Locate the centre of gravity of the resultant flat body.
Ans. COM of resulting portion lies at R/6 from the centre of the original disc in a direction opposite to the centre of the cut out portion.
- The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds, (i) What is its angular acceleration (assume the acceleration to be uniform) (ii) How many revolutions does the wheel make during this time?
- A metre stick is balanced on a knife edge at its centre. When two coins, each of mass 5 g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm, what is the mass of the metre stick?
Ans. m=66.0 g
- A 3m long ladder weighting 20 kg leans on a frictionless wall. Its feet rest on the floor 1 m from the wall as shown in figure. Find the reaction forces F1 and F2 of the wall and the floor.
If F2 makes an angle with the horizontal then
- Calculate the ratio of radii of gyration of a circular ring and a disc of the same radius with respect to the axis passing through their centres and perpendicular to their planes.
- An automobile moves on a road with a speed of 54 km h-1. The radius of its wheels is 0.35 m. What is the average negative torque transmitted by its brakes to a wheel if the vehicle is brought to rest in 15s? The moment of inertia of the wheel about the axis of rotation is 3 kg m2.
- A rod of length L and mass M is hinged at point 0. A small bullet of mass m hits the rod, as shown in figure. The bullet get embedded in the rod. Find the angular velocity of the system just after the impact.
Ans. Using conservation of angular momentum
M V L=l
Or M V L=
- A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously,with initial angular speed equal to 10. Which of the two will start to roll earlier? The coefficient of kinetic friction is pk = 0.2-1π rad s
Ans. The disc begins to roll earlier than the ring.