HC Verma solutions for Concepts of Physics Vol. 1 [English] Class 11 and 12 chapter 10 - Rotational Mechanics [Latest edition]

Can an object be in pure translation as well as in pure rotation?

A simple pendulum is a point mass suspended by a light thread from a fixed point. The particle is displaced towards one side and then released. It makes small oscillations. Is the motion of such a simple pendulum a pure rotation? If yes, where is the axis of rotation?

In a rotating body, \[\alpha = \alpha r\text{ and }\nu = \omega r.\] Thus \[\frac{\alpha}{\alpha} = \frac{\nu}{\omega}.\] Can you use the theorems of ration and proportion studied in algebra so as to write \[\frac{\alpha + \alpha}{\alpha - \alpha} = \frac{\nu + \omega}{\nu - \omega}\]

A ball is whirled in a circle by attaching it to a fixed point with a string. Is there an angular rotation of the ball about its centre? If yes, is this angular velocity equal to the angular velocity of the ball about the fixed point?

The moon rotates about the earth in such a way that only one hemisphere of the moon faces the earth (see the following figure). Can we ever see the "other face" of the moon from the earth? Can a person on the moon ever see all the faces of the earth?

The torque of the weight of any body about any vertical axis is zero. If it always correct?

The torque of a force \[\overrightarrow F \] about a point is defined as \[\overrightarrow\Gamma  =  \overrightarrow r  \times  \overrightarrow F.\] Suppose \[\overrightarrow r, \overrightarrow F\] and \[\overrightarrow \Gamma\] are all nonzero. Is \[r \times  \overrightarrow\Gamma || \overrightarrow F\] always true? Is it ever true?

A heavy particle of mass m falls freely near the earth's surface. What is the torque acting on this particle about a point 50 cm east to the line of motion? Does this torque produce any angular acceleration in the particle?

If several forces act on a particle, the total torque on the particle may be obtained by first finding the resultant force and then taking torque of this resultant. Prove this. Is this result valid for the forces acting on different particles of a body in such a way that their lines of action intersect at a common point?

If the sum of all the forces acting on a body is zero, is it necessarily in equilibrium? If the sum of all the forces on a particle is zero, is it necessarily in equilibrium?

If the angular momentum of a body is found to be zero about a point, is it necessary that it will also be zero about a different point?

If the resultant torque of all the forces acting on a body is zero about a point, is it necessary that it will be zero about any other point?

A body is in translational equilibrium under the action of coplanar forces. If the torque of these forces is zero about a point, is it necessary that it will also be zero about any other point?

A rectangular brick is kept on a table with a part of its length projecting out. It remains at rest if the length projected is slightly less than half the total length but it falls down if the length projected is slightly more than half the total length. Give reason.

When a fat person tries to touch his toes, keeping the legs straight, he generally falls. Explain with reference to the following figure.

A ladder is resting with one end on a vertical wall and the other end on a horizontal floor. If it more likely to slip when a man stands near the bottom or near the top?

When a body is weighed on an ordinary balance we demand that the arum should be horizontal if the weights on the two pans are equal. Suppose equal weights are put on the two pans, the arm is kept at an angle with the horizontal and released. Is the torque of the two weights about the middle point (point of support) zero? Is the total torque zero? If so, why does the arm rotate and finally become horizontal?

The density of a rod AB continuously increases from A to B. Is it easier to set it in rotation by clamping it at A and applying a perpendicular force at B or by clamping it at B and applying the force at A?

When tall building are constructed on earth, the duration of day-night slightly increases. Is it true?

If the ice at the poles melts and flows towards the equator, how will it affect the duration of day-night?

A hollow sphere, a solid sphere, a disc and a ring all having same mass and radius are rolled down on an inclined plane. If no slipping takes place, which one will take the smallest time to cover a given length?

A sphere rolls on a horizontal surface. If there any point of the sphere which has a vertical velocity?

Let \[\overrightarrow A\] be a unit vector along the axis of rotation of a purely rotating body and \[\overrightarrow B\] be a unit vector along the velocity of a particle P of the body away from the axis. The value of \[\overrightarrow A.\overrightarrow B\] is ____________ .

A body is uniformly rotating about an axis fixed in an inertial frame of reference. Let \[\overrightarrow A\] be a unit vector along the axis of rotation and \[\overrightarrow B\] be the unit vector along the resultant force on a particle P of the body away from the axis. The value of \[\overrightarrow A.\overrightarrow B\] is _________.

A particle moves with a constant velocity parallel to the X-axis. Its angular momentum with respect to the origin ____________.

A body is in pure rotation. The linear speed \[\nu\] of a particle, the distance r of the particle from the axis and the angular velocity \[\omega\] of the body are related as \[\omega = \frac{\nu}{r}.\] Thus

The following figure shows a small wheel fixed coaxially on a bigger one of double the radius. The system rotates about the common axis. The strings supporting A and B do not slip on the wheels. If x and y be the distance travelled by A and B in the same time interval, then _________ .

A body is rotating uniformly about a vertical axis fixed in an inertial frame. The resultant force on a particle of the body not on the axis is _________ .

A body is rotating nonuniformity about a vertical axis fixed in an inertial frame. The resultant force on a particle of the body not on the axis is _________ .

Let \[\overrightarrow F\] be a force acting on a particle having position vector \[\overrightarrow r.\] Let \[\overrightarrow\Gamma\] be the torque of this force about the origin, then __________ .

One end of a uniform rod of mass m and length l is clamped. The rod lies on a smooth horizontal surface and rotates on it about the clamped end at a uniform angular velocity \[\omega\] The force exerted by the clamp on the rod has a horizontal component

A uniform rod is kept vertically on a horizontal smooth surface at a point O. If it is rotated slightly and released, it falls down on the horizontal surface. The lower end will remain _________ .

A circular disc A of radius r is made from an iron plate of thickness t and another circular disc B of radius 4r is made from an iron plate of thickness t/4. The relation between the moments of inertia I_A and I_B is __________ .

Equal torques act on the disc A and B of the previous problem, initially both being at rest. At a later instant, the linear speeds of a point on the rim of A and another point on the rim of B are \[\nu_A\] and \[\nu_B\] respectively. We have

A closed cylindrical tube containing some water (not filling the entire tube) lies in a horizontal plane. If the tube is rotated about a perpendicular bisector, the moment of inertia of water about the axis __________ .

The moment of inertia of a uniform semicircular wire of mass M and radius r about a line perpendicular to the plane of the wire through the centre is ___________ .

Let I₁ an I₂ be the moments of inertia of two bodies of identical geometrical shape, the first made of aluminium and the second of iron.

A body having its centre of mass at the origin has three of its particles at (a,0,0), (0,a,0), (0,0,a). The moments of inertia of the body about the X and Y axes are 0⋅20 kg-m² each. The moment of inertia about the Z-axis

A cubical block of mass M and edge a slides down a rough inclined plane of inclination θ with a uniform velocity. The torque of the normal force on the block about its centre has a magnitude

A thin circular ring of mass M and radius r is rotating about its axis with an angular speed ω. Two particles having mass m each are now attached at diametrically opposite points. The angular speed of the ring will become

A person sitting firmly over a rotating stool has his arms stretched. If he folds his arms, his angular momentum about the axis of rotation ___________ .

The centre of a wheel rolling on a plane surface moves with a speed \[\nu_0\] A particle on the rim of the wheel at the same level as the centre will be moving at speed ___________ .

A wheel of radius 20 cm is pushed to move it on a rough horizontal surface. If is found to move through a distance of 60 cm on the road during the time it completes one revolution about the centre. Assume that the linear and the angular accelerations are uniform. The frictional force acting on the wheel by the surface is ______________________ .

The angular velocity of the engine (and hence of the wheel) of a scooter is proportional to the petrol input per second. The scooter is moving on a frictionless road with uniform velocity. If the petrol input is increased by 10%, the linear velocity of the scooter is increased by ___________ .

A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top of a smooth incline and released. Least time will be taken in reaching the bottom by _________ .

A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top on an incline and released. The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling. Least time will be taken in reaching the bottom by ___________ .

In the previous question, the smallest kinetic energy at
the bottom of the incline will be achieved by ___________ .

A string of negligible thickness is wrapped several times around a cylinder kept on a rough horizontal surface. A man standing at a distance l from the cylinder holds one end of the string and pulls the cylinder towards him (see the following figure). There is no slipping anywhere. The length of the string passed through the hand of the man while the cylinder reaches his hands is _________ .

The axis of rotation of a purely rotating body

(a) must pass through the centre of mass

(b) may pass through the centre of mass

(c) must pass through a particle of the body

(d) may pass through a particle of the body.

Consider the following two equations

(A) \[L = I  \omega\]

(B) \[\frac{dL}{dt} = \Gamma\]

In noninertial frames _______________ .

A particle moves on a straight line with a uniform velocity. Its angular momentum __________ .

(a) is always zero

(b) is zero about a point on the straight line

(c) is not zero about a point away from the straight line

(d) about any given point remains constant.

If there is no external force acting on a nonrigid body, which of the following quantities must remain constant?

(a) angular momentum

(b) linear momentum

(c) kinetic energy

(d) moment of inertia.

Let I_A and I_B be moments of inertia of a body about two axes A and B respectively. The axis A passes through the centre of mass of the body but B does not. 

A sphere is rotating about a diameter.

The density of a rod gradually decreases from one end to the other. It is pivoted at an end so that it can move about a vertical axis though the pivot. A horizontal force F is applied on the free end in a direction perpendicular to the rod. The quantities, that do not depend on which end of the rod is pivoted, are ________________ .

Consider a wheel of a bicycle rolling on a level road at a linear speed \[\nu_0\] (see the following figure)

(a) the speed of the particle A is zero

(b) the speed of B, C and D are all equal to \[v_0\]

(c) the speed of C is 2 \[v_0\]

(d) the speed of B is greater than the speed of O.

Two uniform solid spheres having unequal masses and unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping, ___________ .

A hollow sphere and a solid sphere having same mss and same radii are rolled down a rough inclined plane.

A sphere cannot roll on

In rear-wheel drive cars, the engine rotates the rear wheels and the front wheels rotate only because the car moves. If such a car accelerates on a horizontal road the friction

(a) on the rear wheels is in the forward direction

(b) on the front wheels is in the backward direction

(c) on the rear wheels has larger magnitude than the friction on the front wheels

(d) on the car is in the backward direction.

A sphere can roll on a surface inclined at an angle θ if the friction coefficient is more than \[\frac{2}{7}g \tan\theta.\] Suppose the friction coefficient is \[\frac{1}{7}g\ tan\theta.\] If a sphere is released from rest on the incline, _____________ .

A sphere is rolled on a rough horizontal surface. If gradually slows down and stops. The force of friction tries to

(a) decrease the linear velocity

(b) increase the angular velocity

(c) increase the linear momentum

(d) decrease the angular velocity.

The following figure shows a smooth inclined plane fixed in a car accelerating on a horizontal road. The angle of incline θ is related to the acceleration a of the car as a = g tanθ. If the sphere is set in pure rolling on the incline, _____________.

A wheel is making revolutions about its axis with uniform angular acceleration. Starting from rest, it reaches 100 rev/sec in 4 seconds. Find the angular acceleration. Find the angle rotated during these four seconds.

A wheel rotating with uniform angular acceleration covers 50 revolutions in the first five seconds after the start. Find the angular acceleration and the angular velocity at the end of five seconds.

A wheel starting from rest is uniformly accelerated at 4 rad/s² for 10 seconds. It is allowed to rotate uniformly for the next 10 seconds and is finally brought to rest in the next 10 seconds. Find the total angle rotated by the wheel.

A body rotates about a fixed axis with an angular acceleration of one radian/second. Through what angle does it rotate during the time in which its angular velocity increases from 5 rad/s to 15 rad/s.

Find the angular velocity of a body rotating with an acceleration of 2 rev/s² as it completes the 5th revolution after the start.

A disc of radius 10 cm is rotating about its axis at an angular speed of 20 rad/s. Find the linear speed of a point on the rim.

A disc of radius 10 cm is rotating about its axis at an angular speed of 20 rad/s. Find the linear speed of the middle point of a radius.

A disc rotates about its axis with a constant angular acceleration of 4 rad/s². Find the radial and tangential accelerations of a particle at a distance of 1 cm from the axis at the end of the first second after the disc starts rotating.

A block hangs from a string wrapped on a disc of radius 20 cm free to rotate about its axis which is fixed in a horizontal position. If the angular speed of the disc is 10 rad/s at some instant, with what speed is the block going down at that instant?

Three particles, each of mass 200 g, are kept at the corners of an equilateral triangle of side 10 cm. Find the moment of inertial of the system about an axis joining two of the particles.

Three particles, each of mass 200 g, are kept at the corners of an equilateral triangle of side 10 cm. Find the moment of inertial of the system about an axis passing through one of the particles and perpendicular to the plane of the particles.

Particles of masses 1 g, 2 g, 3 g, .........., 100 g are kept at the marks 1 cm, 2 cm, 3 cm, ..........., 100 cm respectively on a metre scale. Find the moment of inertia of the system of particles about a perpendicular bisector of the metre scale.

Find the moment of inertia of a pair of spheres, each having a mass mass m and radius r, kept in contact about the tangent passing through the point of contact.

The moment of inertia of a uniform rod of mass 0⋅50 kg and length 1 m is 0⋅10 kg-m²about a line perpendicular to the rod. Find the distance of this line from the middle point of the rod.

Find the radius of gyration of circular ring of radius r about a line perpendicular to the plane of the ring and passing through one of its particles.

The radius of gyration of a uniform disc about a line perpendicular to the disc equals its radius. Find the distance of the line from the centre.

Find the moment of inertia of a uniform square plate of mass m and edge a about one of its diagonals.

The surface density (mass/area) of a circular disc of radius a depends on the distance from the centre as [rholeft( r right) = A + Br.] Find its moment of inertia about the line perpendicular to the plane of the disc thorough its centre.

A particle of mass m is projected with a speed u at an angle θ with the horizontal. Find the torque of the weight of the particle about the point of projection when the particle is at the highest point.

A simple pendulum of length l is pulled aside to make an angle θ with the vertical. Find the magnitude of the torque of the weight ω of the bob about the point of suspension. When is the torque zero?

When a force of 6⋅0 N is exerted at 30° to a wrench at a distance of 8 cm from the nut it is just able to loosen the nut. What force F would be sufficient to loosen it if it acts perpendicularly to the wrench at 16 cm from the nut?

Calculate the total torque acting on the body shown in the following figure about the point O.

A cubical block of mass m and edge a slides down a rough inclined plane of inclination θ with a uniform speed. Find the torque of the normal force acting on the block about its centre.

A rod of mass m and length L, lying horizontally, is free to rotate about a vertical axis through its centre. A horizontal force of constant magnitude F acts on the rod at a distance of L/4 from the centre. The force is always perpendicular to the rod. Find the angle rotated by the rod during the time t after the motion starts.

A square plate of mass 120 g and edge 5⋅0 cm rotates about one of the edges. If it has a uniform angular acceleration of 0⋅2 rad/s² what torque acts on the plate?

Calculate the torque on the square plate of the previous problem if it rotates about a diagonal with the same angular acceleration.

A flywheel of moment of inertia 5⋅0 kg-m² is rotated at a speed of 60 rad/s. Because of the friction at the axle it comes to rest in 5⋅0 minutes. Find (a) the average torque of the friction (b) the total work done by the friction and (c) the angular momentum of the wheel 1 minute before it stops rotating.

Because of the friction between the water in oceans with the earth's surface the rotational kinetic energy of the earth is continuously decreasing. If the earth's angular speed decreases by 0⋅0016 rad/day in 100 years find the average torque of the friction on the earth. Radius of the earth is 6400 km and its mass is 6⋅0 × 10²⁴ kg.

A wheel rotating at a speed of 600 rpm (revolutions per minute) about its axis is brought to rest by applying a constant torque for 10 seconds. Find the angular deceleration and the angular velocity 5 seconds after the application of the torque.

A wheel of mass 10 kg and radius 20 cm is rotating at an angular speed of 100 rev/min when the motor is turned off. Neglecting the friction at the axle, calculate the force that must be applied tangentially to the wheel to bring it to rest in 10 revolutions.

A cylinder rotating at an angular speed of 50 rev/s is brought in contact with an identical stationary cylinder. Because of the kinetic friction, torques act on the two cylinders accelerating the stationary one and decelerating the moving one. If the common magnitude of the acceleration and deceleration be one revolution per second square, how long will it take before the two cylinders have equal angular speed?

A body rotating at 20 rad/s is acted upon by a constant torque providing it a deceleration of 2 rad/s². At what time will the body have kinetic energy same as the initial value if the torque continues to act?

A light rod of length 1 m is pivoted at its centre and two masses of 5 kg and 2 kg are hung from the ends as shown in the following figure. Find the initial angular acceleration of the rod assuming that it was horizontal in the beginning.

Suppose the rod in the previous problem has a mass of 1 kg distributed uniformly over its length.

(a) Find the initial angular acceleration of the rod.

(b) Find the tension in the supports to the blocks of mass 2 kg and 5 kg.

The following figure shows two blocks of mass m and M connected by a string passing over a pulley. The horizontal table over which the mass m slides is smooth. The pulley has a radius r and moment of inertia I about its axis and it can freely rotate about this axis. Find the acceleration of the mass M assuming that the string does not slip on the pulley.

A string is wrapped on a wheel of moment of inertia 0⋅20 kg-m² and radius 10 cm and goes through a light pulley to support a block of mass 2⋅0 kg as shown in the following figure. Find the acceleration of the block.

Suppose the smaller pulley of the previous problem has its radius 5⋅0 cm and moment of inertia 0⋅10 kg-m². Find the tension in the part of the string joining the pulleys.

The pulleys shown in the following figure are identical, each having a radius R and moment of inertia I. Find the acceleration of the block M.

The descending pulley shown in the following figure has a radius 20 cm and moment of inertia 0⋅20 kg-m². The fixed pulley is light and the horizontal plane frictionless. Find the acceleration of the block if its mass is 1⋅0 kg.

The pulley shown in the following figure has a radius 10 cm and moment of inertia 0⋅5 kg-m²about its axis. Assuming the inclined planes to be frictionless, calculate the acceleration of the 4⋅0 kg block.

Solve the previous problem if the friction coefficient between the 2⋅0 kg block and the plane below it is 0⋅5 and the plane below the 4⋅0 kg block is frictionless.

A uniform metre stick of mass 200 g is suspended from the ceiling thorough two vertical strings of equal lengths fixed at the ends. A small object of mass 20 g is placed on the stick at a distance of 70 cm from the left end. Find the tensions in the two strings.

A uniform ladder of length 10⋅0 m and mass 16⋅0 kg is resting against a vertical wall making an angle of 37° with it. The vertical wall is frictionless but the ground is rough. An electrician weighing 60⋅0 kg climbs up the ladder. If he stays on the ladder at a point 8⋅00 m from the lower end, what will be the normal force and the force of friction on the ladder by the ground? What should be the minimum coefficient of friction for the electrician to work safely?

Suppose the friction coefficient between the ground and
the ladder of the previous problem is 0.540. Find the
maximum weight of a mechanic who could go up and do
the work from the same position of the ladder.

A 6⋅5 m long ladder rests against a vertical wall reaching a height of 6⋅0 m. A 60 kg man stands half way up the ladder.

1. Find the torque of the force exerted by the man on the ladder about the upper end of the ladder.
2. Assuming the weight of the ladder to be negligible as compared to the man and assuming the wall to be smooth, find the force exerted by the ground on the ladder.

The door of an almirah is 6 ft high, 1⋅5 ft wide and weighs 8 kg. The door is supported by two hinges situated at a distance of 1 ft from the ends. If the magnitudes of the forces exerted by the hinges on the door are equal, find this magnitude.

A uniform rod of length L rests against a smooth roller as shown in the following figure. Find the friction coefficient between the ground and the lower end if the minimum angle that the rod can make the horizontal is θ.

A uniform rod of mass 300 g and length 50 cm rotates at a uniform angular speed of 2 rad/s about an axis perpendicular to the rod through an end. Calculate (a) the angular momentum of the rod about the axis of rotation, (b) the speed of the centre of the rod and (c) its kinetic energy.

A uniform square plate of mass 2⋅0 kg and edge 10 cm rotates about one of its diagonals under the action of a constant torque of 0⋅10 N-m. Calculate the angular momentum and the kinetic energy of the plate at the end of the fifth second after the start.

Calculate the ratio of the angular momentum of the earth about its axis due to its spinning motion to that about the sun due to its orbital motion. Radius of the earth = 6400 km and radius of the orbit of the earth about the sun = 1⋅5 × 10⁸ km.

Two particles of masses m₁ and m₂ are joined by a light rigid rod of length r. The system rotates at an angular speed ω about an axis through the centre of mass of the system and perpendicular to the rod. Show that the angular momentum of the system is \[L = \mu r^2 \omega\] where \[\mu\] is the reduced mass of the system defined as \[\mu = \frac{m_1 + m_2}{m_1 + m_2}\]

A dumb-bell consists of two identical small balls of mass 1/2 kg each connected to the two ends of a 50 cm long light rod. The dumb-bell is rotating about a fixed axis thorough the centre of the rod and perpendicular to it at an angular speed of 10 rad/s. An impulsive force of average magnitude 5⋅0 N acts on one of the masses in the direction of its velocity for 0⋅10 s. Find the new angular velocity of the system.

A wheel of moment of inertia 0⋅500 kg-m² and radius 20⋅0 cm is rotating about its axis at an angular speed of 20⋅0 rad/s. It picks up a stationary particle of mass 200 g at its edge. Find the new angular speed of the wheel.

A diver having a moment of inertia of 6⋅0 kg-m² about an axis thorough its centre of mass rotates at an angular speed of 2 rad/s about this axis. If he folds his hands and feet to decrease the moment of inertia to 5⋅0 kg-m², what will be the new angular speed?

A boy is seated in a revolving chair revolving at an angular speed of 120 revolutions per minute. Two heavy balls form part of the revolving system and the boy can pull the balls closer to himself or may push them apart. If by pulling the balls closer, the boy decreases the moment of inertia of the system from 6 kg-m² to 2 kg-m², what will be the new angular speed?

A boy is standing on a platform which is free to rotate about its axis. The boy holds an open umbrella in his hands. The axis of the umbrella coincides with that of the platform. The moment of inertia of "the platform plus the boy system" is 3⋅0 × 10⁻³ kg-m² and that of the umbrella is 2⋅0 × 10⁻³ kg-m². The boy starts spinning the umbrella about the axis at an angular speed of 2⋅0 rev/s with respect to himself. Find the angular velocity imparted to the platform.

A wheel of moment of inertia 0⋅10 kg-m² is rotating about a shaft at an angular speed of 160 rev/minute. A second wheel is set into rotation at 300 rev/minute and is coupled to the same shaft so that both the wheels finally rotate with a common angular speed of 200 rev/minute. Find the moment of inertia of the second wheel.

A kid of mass M stands at the edge of a platform of radius R which can be freely rotated about its axis. The moment of inertia of the platform is I. The system is at rest when a friend throws a ball of mass m and the kid catches it. If the velocity of the ball is \[\nu\] horizontally along the tangent to the edge of the platform when it was caught by the kid, find the angular speed of the platform after the event.

Suppose the platform of the previous problem is brought to rest with the ball in the hand of the kid standing on the rim. The kid throws the ball horizontally to his friend in a direction tangential to the rim with a speed \[\nu\] as seen by his friend. Find the angular velocity with which the platform will start rotating.

Suppose the platform with the kid in the previous problem is rotating in anticlockwise direction at an angular speed ω. The kid starts walking along the rim with a speed \[\nu\] relative to the platform also in the anticlockwise direction. Find the new angular speed of the platform.

A uniform rod of mass m and length l is struck at an end by a force F perpendicular to the rod for a short time interval t. Calculate (a) the speed of the centre of mass, (b) the angular speed of the rod about the centre of mass, (c) the kinetic energy of the rod and (d) the angular momentum of the rod about the centre of mass after the force has stopped to act. Assume that t is so small that the rod does not appreciably change its direction while the force acts.

A uniform rod of length L lies on a smooth horizontal table. A particle moving on the table strikes the rod perpendicularly at an end and stops. Find the distance travelled by the centre of the rod by the time it turns through a right angle. Show that if the mass of the rod is four times that of the particle, the collision is elastic.

Suppose the particle of the previous problem has a mass m and a speed \[\nu\] before the collision and it sticks to the rod after the collision. The rod has a mass M. (a) Find the velocity of the centre of mass C of the system constituting "the rod plus the particle". (b) Find the velocity of the particle with respect to C before the collision. (c) Find the velocity of the rod with respect to C before the collision. (d) Find the angular momentum of the particle and of the rod about the centre of mass C before the collision. (e) Find the moment of inertia of the system about the vertical axis through the centre of mass C after the collision. (f) Find the velocity of the centre of mass C and the angular velocity of the system about the centre of mass after the collision.

Two small balls A and B, each of mass m, are joined rigidly by a light horizontal rod of length L. The rod is clamped at the centre in such a way that it can rotate freely about a vertical axis through its centre. The system is rotated with an angular speed ω about the axis. A particle P of mass m kept at rest sticks to the ball A as the ball collides with it. Find the new angular speed of the rod.

Two small balls A and B, each of mass m, are joined rigidly to the ends of a light rod of length L (see the following figure). The system translates on a frictionless horizontal surface with a velocity \[\nu_0\] in a direction perpendicular to the rod. A particle P of mass m kept at rest on the surface sticks to the ball A as the ball collides with it. Find
(a) the linear speeds of the balls A and B after the collision, (b) the velocity of the centre of mass C of the system A + B + P and (c) the angular speed of the system about C after the collision.

[Hint : The light rod will exert a force on the ball B
only along its length.]

Suppose the rod with the balls A and B of the previous problem is clamped at the centre in such a way that it can rotate freely about a horizontal axis through the clamp. The system is kept at rest in the horizontal position. A particle P of the same mass m is dropped from a height h on the ball B. The particle collides with B and sticks to it. (a) Find the angular momentum and the angular speed of the system just after the collision. (b) What should be the minimum value of h so that the system makes a full rotation after the collision.

Two blocks of masses 400 g and 200 g are connected through a light string going over a pulley which is free to rotate about its axis. The pulley has a moment of inertia \[1 \cdot 6 \times  {10}^{- 4}   kg -  m^2\] and a radius 2⋅0 cm, Find (a) the kinetic energy of the system as the 400 g block falls through 50 cm, (b) the speed of the blocks at this instant.

The pulley shown in the following figure has a radius of 20 cm and moment of inertia 0⋅2 kg-m². The string going over it is attached at one end to a vertical spring of spring constant 50 N/m fixed from below, and supports a 1 kg mass at the other end. The system is released from rest with the spring at its natural length. Find the speed of the block when it has descended through 10 cm. Take g = 10 m/s².

A metre stick is held vertically with one end on a rough horizontal floor. It is gently allowed to fall on the floor. Assuming that the end at the floor does not slip, find the angular speed of the rod when it hits the floor.

A metre stick weighing 240 g is pivoted at its upper end in such a way that it can freely rotate in a vertical place through this end (see the following figure). A particle of mass 100 g is attached to the upper end of the stick through a light string of length 1 m. Initially, the rod is kept vertical and the string horizontal when the system is released from rest. The particle collides with the lower end of the stick and sticks there. Find the maximum angle through which the stick will rise.

A uniform rod pivoted at its upper end hangs vertically. It is displaced through an angle of 60° and then released. Find the magnitude of the force acting on a particle of mass dm at the tip of the rod when the rod makes an angle of 37° with the vertical.

A cylinder rolls on a horizontal place surface. If the speed of the centre is 25 m/s, what is the speed of the highest point?

A sphere of mass m rolls on a plane surface. Find its kinetic energy at an instant when its centre moves with speed \[\nu.\]

A string is wrapped over the edge of a uniform disc and the free end is fixed with the ceiling. The disc moves down, unwinding the string. Find the downward acceleration of the disc.

A small spherical ball is released from a point at a height h on a rough track shown in the following figure. Assuming that it does not slip anywhere, find its linear speed when it rolls on the horizontal part of the track.

A small disc is set rolling with a speed \[\nu\] on the horizontal part of the track of the previous problem from right to left. To what height will it climb up the curved part?

A sphere starts rolling down an incline of inclination θ. Find the speed of its centre when it has covered a distance l.

A hollow sphere is released from the top of an inclined plane of inclination θ. (a) What should be the minimum coefficient of friction between the sphere and the plane to prevent sliding? (b) Find the kinetic energy of the ball as it moves down a length l on the incline if the friction coefficient is half the value calculated in part (a).

A solid sphere of mass m is released from rest from the rim of a hemispherical cup so that it rolls along the surface. If the rim of the hemisphere is kept horizontal, find the normal force exerted by the cup on the ball when the ball reaches the bottom of the cup.

The following figure shows a rough track, a portion of which is in the form of a cylinder of radius R. With what minimum linear speed should a sphere of radius r be set rolling on the horizontal part so that it completely goes round the circle on the cylindrical part.

The following figure shows a small spherical ball of mass m rolling down the loop track. The ball is released on the linear portion at a vertical height H from the lowest point. The circular part shown has a radius R.
(a) Find the kinetic energy of the ball when it is at a point A where the radius makes an angle θ with the horizontal.
(b) Find the radial and the tangential accelerations of the centre when the ball is at A.
(c) Find the normal force and the frictional force acting on the if ball if H = 60 cm, R = 10 cm, θ = 0 and m = 70 g.

A thin spherical shell of radius R lying on a rough horizontal surface is hit sharply and horizontally by a cue. Where should it be hit so that the shell does not slip on the surface?

A uniform wheel of radius R is set into rotation about its axis at an angular speed ω. This rotating wheel is now placed on a rough horizontal surface with its axis horizontal. Because of friction at the contact, the wheel accelerates forward and its rotation decelerates till the wheel starts pure rolling on the surface. Find the linear speed of the wheel after it starts pure rolling.

A thin spherical shell lying on a rough horizontal surface is hits by a cue in such a way that the line of action passes through the centre of the shell. As a result, the shell starts moving with a linear speed \[\nu\] without any initial angular velocity. Find the linear speed of the shell after it starts pure rolling on the surface.

A hollow sphere of radius R lies on a smooth horizontal surface. It is pulled by a horizontal force acting tangentially from the highest point. Find the distance travelled by the sphere during the time it makes one full rotation.

A solid sphere of mass 0⋅50 kg is kept on a horizontal surface. The coefficient of static friction between the surfaces in contact is 2/7. What maximum force can be applied at the highest point in the horizontal direction so that the sphere does not slip on the surface?

A solid sphere is set into motion on a rough horizontal surface with a linear speed ν in the forward direction and an angular speed ν/R in the anticlockwise directions as shown in the following figure. Find the linear speed of the sphere (a) when it stops rotating and (b) when slipping finally ceases and pure rolling starts.

A solid sphere rolling on a rough horizontal surface with a linear speed ν collides elastically with a fixed, smooth, vertical wall. Find the speed of the sphere after it has started pure rolling in the backward direction.