Rigid Body Dynamics | Visual Physics for IIT JEE / NEET
Introduction

Motion of everyday objects is a combination of Translational and Rotational Motion. So how do we represent such a motion ? Can we say that the combined motion is the vector sum of the Translational motion of their CoM and Rotational motion wrt CoM ?

Pure Rolling - 1

Pure Rolling is defined as rolling without slipping. In pure rolling, the point of contact has zero speed i.e it is stationary wrt ground. But how can the point of contact have zero speed ? How can it be stationary wrt ground ?

Q1

A disc of radius R rolls without slipping at speed v along positive x-axis. velocity of point P at the instant shown in fig. is

Q2

Figure shows the velocities of the plank and cylinder in ground frames and the cylinder is performing pure rolling on the plank, velocity of point A would be (a) 2vC (b) 2vC + vP (c) 2vC ? vP (d) 2( vC ?vP )

Q3

A ball of radius R is rolling without slipping on horizontal rails with velocity v. Its center of mass is at a distance d above the rails. Find the relation between the linear and angular velocity of the ball.

Pure Rolling - 2

Relationship between linear and angular acceleration of a body doing pure rolling.

Q4

A ball of radius R rolls on horizontal ground with linear acceleration a and angular acceleration a as shown in figure. The magnitude of acceleration point P as shown in figure at an instant when its linear velocity is v and angular velocity is w will be

Inst Axis of Rotation

What is Instantaneous Axis of rotation ? How can an axis of rotation be an axis just for a moment ? Does the instantaneous axis have to pass through the body ?

Q5

A small ball strikes at one end of a stationary uniform frictionless rod of mass m and length l which is free to rotate in a gravity free space. The impact is elastic. Instantaneous axis of rotaion of the rod will pass through a) its centre of mass. b) the centre of mass of the rod plus ball c) the point of impact of the ball on the rod d) the point 2l/3 from the striking end

Q6

A disk is movnig towards +ve x-axis with a velocity vc and rotates clockwise with angular speed w as shown in fig. such that vc>wR. The instantaneous axis of rotation will be

Kinetic Energy

Kinetic Energy of a body having both translational and rotational motion.

Q7

If a spherical ball rolls on a table without slipping, the fraction of its total energy associated with rotation is (a) 3/5 (b) 2/7 (c) 2/5 (d) 3/7

Ang Momentum

Derivation and Understanding Angular Momentum. Does Angular Momentum depend on the point of reference ? Does Moment of Inertia depend on the point of reference ? Does Angular velocity depend on the point of reference ?

Q8

A disc of mass m and radius R moves in the x-y plane as shown in figure. The angular momentum of the disc about the origin O at the instant shown will be

Q9

Two points of a rod move with velocities 3v and v perpendicular to the rod and in the same direction separated by a distance r. The angular velocity of the rod is

Torque

Net Torque on a body is equal to the rate of change of its Angular Momentum. It is the rotational counterpart of Newton?s 2nd Law. Does Torque depend on the Point of Reference ? Is the above equation valid in all Frames of Reference ? Is the above equation valid in the frame of CoM, even if it is a non-inertial frame of reference ?

Q10

A uniform disc is moving on a horizontal frictionless surface such that the velocity of its CoM is vo and its angular speed is wo. It is hooked at a rigid point P and rotates without bouncing. Find its angular speed after the impact.

Q11

A uniform smooth rod placed on a smooth horizontal floor is hit by a particle moving on the floor, at a distance L/4 from one end. Find the distance travelled by the centre of the rod after the collision when it has completed n revolutions. [ e ? 0 and L is the length of the rod ]

Q12

A sphere is released on a smooth inclined plane from the top. When it moves down, its angular momentum is a) conserved about every point b) conserved about the point of contact only c) conserved about the center of the sphere only d) conserved about any point on a line parallel to the inclined plane and passing through the center of the ball

Q13

Given that a ball roll down a rough incline plane without slipping, is there a point wrt which the angular momentum of the ball is conserved ?

Equilibrium

Conditions for Equilibrium of Rigid Bodies. Is a body rotating with constant angular velocity in Equilibrium ? Is a body moving with constant velocity and rotating with constant angular velocity in Equilibrium ?

Q14

A uniform rod, of length L and mass M, is at rest on two supports. A uniform block, with mass m is at rest on the rod, with its center a distance L/3 from the rod?s left end. What are the normal reactions from the supports on rod ?

Q15

A ladder of length L and mass M leans against a frictionless wall. The ladders upper end is at height h above the ground on which the lower end rests (the ground is not frictionless). A man of mass m climbs the ladder. What are the forces on the ladder from the wall and the ground a) when the man is mid-way between the ladder b) when is the man more likely to fall, near the bottom or top c) find minimum m so that the ladder does not slip

Toppling

How does a block topple ? Does Normal Reaction always act uniformly over the surface in contact ? Or can it be unevenly distributed ?

Q16

The block of dimensions l by h is lying on an incline of angle q. Find the maximum q for which it does not topple. Coefficient of friction between incline and block is m.

Dyn of Rolling

What will be the direction of Friction acting on a Rolling object ?

Q17

Figure shows a spool of mass m and radius R which is pulled by a constant horizontal force F on a rough horizontal surface. The radius of the drum is r on which a string tightly wound. The moment of inertia of the spool about its center of mass is I = mk2.

Q18

A uniform cylinder of radius R is spinned about its axis to the angular velocity wo and then placed into a corner. The coefficient of friction between the corner walls and the cylinder is m . How many turns will the cylinder accomplish before it stops?

Q19

A spherical ball of mass m and radius R is thrown along a rough horizontal surface so that initially (t = 0) it slide with a linear speed v0 but does not rotate. As it slides, it begins to spin and eventually rolls without slipping. How long does it take to begin rolling without slipping?

Q20

Why does a rolling ball slow down ?

Rolling on Incline

Dynamics and Energy considerations of an object rolling on an Incline plane.

Q21

A solid sphere, a hollow sphere, a solid cylinder, a hollow cylinder and a ring, all having same mass and radius, are place at the top of an incline and released. The friction coefficient between the object and the incline is not sufficient to allow pure rolling. Which object will reach the bottom of the incline first ?

Q22

A cylinder of mass m is suspended through two strings wrapped around it as shown in figure. Find (a) the tension T in the string and (b) acceleration of the cylinder.

1

At the bottom edge of a smooth wall, an inclined plane is kept at an angle of 450. A uniform rod of length l and mass m rests on the inclined plane against the wall such that it is perpendicular to the incline. a) If the plane is also smooth, which way will the rod slide? b) What is the minimum coefficient of friction necessary so that the rod does not slip on the incline?

2

A car of mass m traveling at speed v moves on a horizontal track. The center of mass of the car describes a circle of radius r. If 2a is the separation between the wheels and h is the height of the center of mass above the ground, show that the limiting speed beyond which the car will overturn is given by

3

Determine the minimum co-efficient of friction between a thin rod and a floor at which a person can slowly lift the rod from the floor, without slipping, to the vertical position applying at its end a force always perpendicular to its length.

4

A uniform cylinder at rest on a rough horizontal rug that is pulled out from under it with an acceleration (arug) perpendicular to the axis of the cylinder. Find the force of friction on cylinder, assuming it does not slip.

5

A system of uniform cylinders and planks is shown in figure. All the cylinders are identical and there is no slipping at any contact. Velocity of lower and upper planks is v1 and v2, respectively as shown in figure. Find the angular speeds of the upper and lower cylinders.

6

Two steel balls of masses m1 and m2 are connected by a massless rigid bar of length l. They fall in a horizontal position from a height h on two heavy steel and brass plates as shown. The coefficient of restitution between the balls and steel and brass plates are e1 and e2, respectively. Assuming that the two balls hit the respective plates at the same instant, find the angular velocity w of the bar immediately after impact.

7

A uniform ball of radius r rolls without slipping down from the top of a sphere of radius R. Find the velocity of the ball at the moment it breaks off the sphere. The initial velocity of the ball is negligible.

8

A small solid sphere of radius r rolls down an incline without slipping which ends into a vertical loop of radius R. Find the height above the base so that it just loops the loop.

9

A ball intially rolls without sliding, on a horizontal surface. It ascends a curved track up to height hr on rough surface (without sliding) and hs on smooth surface. Find the value of hr and hs.

10

A ball rolls down (without slipping) from A (starting from rest) then it comes on smoothly joined horizontal section BC and then rolls on to CD to reach D. It rolls back from point D and reaches some height on the rough surface. Mark out the correct statement (s). (a) on BC the ball performs pure rolling (b) ball has rotational motion at highest point on CD (c) h1 > h2 (d) ball will again reach at A

11

A ball of mass m1 is initially rolling without sliding with a velocity u on the horizontal surface of a frictionless wedge as shown in the figure. Wedge has a mass m2. All surfaces are smooth. Wedge has no initial velocity. What will be the maximum height reached by the ball.

12

A solid sphere S and a thin hoop of equal mass m and R are coupled together by a massless road. This assembly is free to roll down the inclined plane without slipping. Determine the the force developed in the rod and the acceleration of the system.

13

A solid sphere of mass m and radius R is placed on a rough horizontal surface. It is stuck by a horizontal cuestick at a height h above the surface. The value of h, so that the sphere performs pure rolling motion immediately after it has been stuck is

14

A billiard ball (of radius R), initially at rest is given a sharp impulse by a cue. The cue is held horizontally at distance h above the central line as shown figure. The ball leaves the cue with a speed vo. It rolls and slide while moving forward and eventually acquires a final speed of 9/7 vo. Show that h = 4/5 R.

15

A uniform object is set in motion with back spin, on a rough horizontal surface, as shown in fig. If given velocity is vo and angular velocity is wo, Find relation between vo and wo so the object can come back. What is the relation if the object is a ring, cylinder or a sphere.

16

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

17

A ball of radius R is rolling on a horizontal surface with velocity vo and angular velocity wo (vo = woR). The sphere collides with a sharp edge on the wall as shown in figure. The coefficient of friction between the sphere and the edge m. Just after the collision the angular velocity of the sphere becomes equal to zero. Find the linear velocity of the sphere just after the collision. (assume slipping occurs while the ball is in contact with the edge and ball does not bounce)

18

A solid ball of mass m and radius R spinning with angular velocity wo falls on a horizontal slab of mass M with rough upper surface (coefficient of friction m) and smooth lower surface. Immediately after collision the normal component of velocity of the ball remains half of its value just before collision and it stops spinning. Find the velocity of the sphere in horizontal direction immediately after impact.

19

A spherical ball of mass m and radius R moving with velocity u strikes elastically with a rigid surface at an angle q to the normal. Assume that slipping occurs while the sphere is in contact with the surface , the frictional coefficient is m ; shows that

20

Find the acceleration of a system consisting of a cylinder of mass m and radius R and a plank of mass M placed an a smooth surface if it is pulled with a force F as shown figure. Given that sufficient friction is present between the cylinder and the plank surface to prevent sliding of cylinder. (pulley and string is massless and there is no friction an the axle of the pulley)

21

A double pulley of mass M, outer radius R and inner radius r is kept on rough surface. A light inextensible string is wound on the inner pulley and is attached to a mass m as shown in figure. There is no slipping between the pulley and string and the pulley and ground. Find acceleration of the block.

22

A solid cylinder of mass M and radius R is rolled up on an incline with the help of a plank of mass 2M. A constant force F is acting on the plank parallel to the incline. There is no slipping at any of the contact. Find the friction force between the plank and cylinder.

23

A solid cylinder is placed on an inclined plane. It is found that the plane can be tilted at an angle q before the cylinder starts to slide. When the cylinder turns on its sides and is allows to roll, it is found that the steepest angle at which cylinder performs pure rolling is f. Find the ratio tan f/ tan q

24

A unform solid cylinder of mass m rests on a rough surface as shown in fig. A thread is wound on the cylinder. The free end of the thread is pulled vertically up with a force F. What is the maximum magnitude of the force F which still does not bring about any sliding of the cylinder, if the coefficient of friction between the cylinder and the plank is equal to m. What is the maximum acceleration of the axis of cylinder?

25

Find the tension in the thread and the linear acceleration of the cylinder up the incline, assuming there is no slipping of the thread over the cylinder and the cylinder over the incline.

26

In the arrangement shown in figure, the block has mass m. The double pulley has mass M and moment of inertia I relative to the axis. Radii of the pulley are R and 2R. (mass of the thread is negligible) Find the acceleration of the block after the system is set free.

27

A ball of radius R is rolling without slipping with linear velocity vo . It encounters a step of height h as shown in the figure. Find the minimum speed of the ball for which it rises up the step. (friction is sufficient to prevent slipping of the ball)

28

A carpet of mass M made of inextensible material is rolled along its length in the form of a cylinder of radius R and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligibly small push is given to it. Calculate the horizontal velocity of the axis of the cylindrical part of the carpet as a function of its radius.

29

A uniform solid cylinder of radius R rolls on a horizontal surface that passes into an inclined plane of inclination q. Find the maximum value of vo which still allows it to roll on the inclined surface without a jump. Assume that the cylinder rolls without sliding.

30

A thin uniform bar lies on a frictionless horizontal surface and is free to move in any way on the surface. Its mass is M and length is l. Two particles, each of mass m are moving on the same surface and towards the bar in a direction perpendicular to the bar, one with a velocity of u1 and the other with u2 , as shown in figure.The particles strike the bar at the same distance r from the center of rod and at same instance of time and stick to the bar on collision. Calculate the loss of kinetic energy of the system in the above collision process.

31

A square plate ABCD of mass m and side l is suspended with the help of two ideal strings P and Q as shown. Determine the acceleration of the corner A of the square just at the moment the string Q is cut.

32

A uniform plank leans against a cylindrical body as shown in fig. The right end of the plank slides at a constant speed v. Find (a) the angular speed w and (b) the angular acceleration a in terms of v, x and R.

33

Two identical cylinders of mass M and radius R are connected by a light rod. The assembly rests at a corner, the vertical wall is smooth and there is sufficient friction on the floor to ensure pure rolling of the cylinder B. The system starts from position q = 0. Find the velocity of the midpoint of the rod when the rod makes an angle q with the vertical.

34

Angular momentum of a particle about a stationary point O varies with time as L = a + bt2 where a and b are constant vectors with a perpendicular to b. The torque t acting on the particle when angle between t and L is 45o is

35

The torque acting on a body about a given point is given by t = A L where A is a constant vector and L is the angular momentum of the body about that point. It follows that a) is perpendicular to L at all instant of time. b) the component of L in the direction of A does not change with time. c) the magnitude of L does not change with time. d) All the above choices are correct.

36

A small mass m is attached inside of the rigid ring of the same mass m and radius R. The ring performs pure rolling on a rough horizontal surface. At the moment the mass m gets into the lowest position, the center of the ring moves with velocity v0. For what value of v0, the ring moves without bouncing?

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