**Rigid Body Dynamics**

- This video discusses the motion of a general object which consists of both translational and rotational motion. Any such motion of a rigid body can be considered as pure translational motion of its centre of mass plus pure rotational motion of rigid body with respect to its centre of mass.
- This video explains the combination of translational and rotational motion through an example of motion of a rolling ring. The motion of any point on the ring can be considered as vector sum of translational and rotational motion. It also explains the corresponding equations of pure rolling.
- A disc of radius R rolls without slipping at speed v along positive x-axis. Velocity of point P at the instant shown in figure is…?
- 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 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.
- This video explains the phenomena of centripetal acceleration with respect to the ground. It also states the equations relating the linear acceleration and rotational acceleration.
- A ball of radius R rolls on horizontal ground with a given linear and angular acceleration. The magnitude of acceleration point P as shown in figure at an instant when its linear and angular velocity is known; will be…?
- This video discusses the method to determine the velocity of any point on a body moving under pure rolling. It also explains the instantaneous axis of rotation and method to determine it for any rigid body.
- A small ball strikes elastically 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. Instantaneous axis of rotation of the rod will pass through which of the following? (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
- A disk is moving towards positive x-axis with a velocity vc and rotates clockwise with angular speed ω as shown in figure such that vc > ωR. The instantaneous axis of rotation will be (a) at point P (b) at point P` (c) inside the sphere (d)outside the sphere
- This video illustrates the derivation of kinetic energy equation for a rigid body moving under the combination of translational and rotational motion.
- If a spherical ball rolls on a table without slipping, the fraction of its total energy associated with rotation is….?
- This video shows the derivation of angular momentum of a rigid body moving under translational plus rotational motion. It also explains the independence of angular momentum on the point of reference.
- A disc of mass m and radius R moves in the x-y plane. The angular momentum of the disc about the origin O at the instant shown in figure will be….?
- 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…?
- This video discusses the concept of torque for a rigid body which is defined as the rate of change of angular momentum. It also explains the concept of pseudo torque for different frame of references.
- A uniform disc is moving on a horizontal frictionless surface such that the velocity of its Centre of Mass and its angular speed is known. It is hooked at a rigid point P and rotates without bouncing as shown in figure. Find its angular speed after the impact.
- 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.
- A sphere is released on a smooth inclined plane from the top. When it moves down, which of the following is correct about angular momentum? It 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.
- Given that a ball roll down a rough incline plane without slipping as shown in figure, is there a point with respect to which the angular momentum of the ball is conserved?
- This video discusses the role of net force along with net torque to define the equilibrium of a rigid body.
- A uniform rod, of length L and mass M, is at rest on two supports as shown in figure. 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?
- A ladder of length L and mass M leans against a frictionless wall. The ladder's 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 mass of the man so that the ladder does not slip.
- This video discusses the conditions required for toppling of an object in relation with the displaced normal reaction and friction.
- The block of given dimensions is lying on an incline of angle θ. Find the maximum angle θ for which it does not topple. Coefficient of friction between incline and block is known.
- This video explains in detail, the dynamics and conditions for pure rolling as well as rolling with slipping.
- 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. Find the friction and its direction.
- A uniform cylinder of radius R is spinned about its axis to a given angular velocity and then placed into a corner. The coefficient of friction between the corner walls and the cylinder is known . How many turns will the cylinder accomplish before it stops?
- 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?
- Why does a rolling ball slow down on its own without any apparent force acting on it?
- This video describes the dynamics of the rolling on an inclined rough plane and the dependence of acceleration and velocity at the bottom on the moment of inertia of the object.
- 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 ?
- 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 when released from rest.
- At the bottom edge of a smooth wall, an inclined plane is kept at an angle. 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?
- 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, find the limiting speed beyond which the car will overturn.
- 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 length of the rod.
- A uniform cylinder at rest on a rough horizontal rug that is pulled out from under it with an acceleration perpendicular to the axis of the cylinder. Find the force of friction on cylinder, assuming it does not slip.
- As 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 in opposite direction. Find the angular speeds of the upper and lower cylinders.
- Two steel balls of masses m1 and m2 are connected by a mass-less rigid bar of length l. They fall in a horizontal position from a height h on two heavy steel and brass plates as shown in figure. 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 of the bar immediately after impact.
- A uniform ball of radius r rolls down without slipping from the top of a sphere of radius R as shown in figure. Find the velocity of the ball at the moment it breaks off the sphere. The initial velocity of the ball is negligible.
- A small solid sphere of radius r rolls down an incline without slipping which ends into a vertical loop of radius R as shown in figure. Find the height above the base so that it just loops the loop.
- A ball initially rolls without sliding, on a horizontal surface. It ascends a curved track up to different heights on rough surface (without sliding) and smooth surface. Find the value of heights it ascends.
- A ball rolls down (without slipping) from A from rest to a smoothly joined horizontal section BC and then rolls on to CD to reach D as shown in figure. 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
- A ball of mass m1 is initially rolling without sliding with a velocity 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.
- A solid sphere S and a thin hoop of equal mass m and R are coupled together by a mass-less road. This assembly is free to roll down the inclined plane without slipping. Determine the force developed in the rod and the acceleration of the system.
- A solid sphere of mass m and radius R is placed on a rough horizontal surface. It is stuck by a horizontal cue-stick 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…?
- 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 given final speed. Show that h = 4/5 R.
- A uniform object is set in motion with back spin, on a rough horizontal surface as shown in figure. If the given velocity and angular velocity is known, Find relation between linear and angular velocity so the object can come back. What is the relation if the object is a ring, cylinder or a sphere?
- A solid sphere rolling on a rough horizontal surface with a linear speed collides elastically with a fixed, smooth, vertical wall. Find the speed of the sphere after it has started pure rolling in the backward direction.
- A ball of radius R is rolling on a horizontal surface with a known linear and angular velocity. The sphere collides with a sharp edge on the wall as shown in figure. The coefficient of friction between the sphere and the edge is known. 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)
- A solid ball of mass m and radius R spinning with a given angular velocity falls on a horizontal slab of mass M with rough upper surface (coefficient of friction given) 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.
- A spherical ball of mass m and radius R moving with velocity u strikes elastically with a rigid surface with coefficient of friction µ at an angle θ to the normal. Assume that slipping occurs while the sphere is in contact with the surface; find relation with angles of incidence and reflection. Also find the change in angular velocity.
- Find the acceleration of a system consisting of a cylinder of mass m and radius R and a plank of mass M placed on a smooth surface if it is pulled with a force F as shown in figure. Given that sufficient friction is present between the cylinder and the plank surface to prevent sliding of cylinder. (Assume pulley and string are mass-less and there is no friction at the axle of the pulley)
- 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.
- 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.
- A solid cylinder is placed on an inclined plane. It is found that the plane can be tilted at an angle θ 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 φ. Find the ratio tan φ / tan θ.
- A uniform solid cylinder of mass m rests on a rough surface with known coefficient of friction as shown in figure. 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 and maximum acceleration of the axis of cylinder?
- For the system shown in figure, 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.
- 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.
- 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)
- 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.
- A uniform solid cylinder of radius R rolls on a horizontal surface that passes into a plane inclined at an angle θ. Find the maximum value of speed vo which still allows it to roll on the inclined surface without a jump. Assume that the cylinder rolls without sliding.
- 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.
- A square plate ABCD of mass m and side l is suspended with the help of two ideal strings P and Q as shown in figure. Determine the acceleration of the corner A of the square just at the moment the string Q is cut.
- A uniform plank leans against a cylindrical body as shown in figure. The right end of the plank slides at a given constant speed. Find the angular speed and the angular acceleration of the plank.
- 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 θ = 0. Find the velocity of the midpoint of the rod when the rod makes an angle θ with the vertical.
- 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 τ acting on the particle when angle between τ and L is 45 degree is ….?
- The torque acting on a body about a given point is given by τ = A x L where A is a constant vector and L is the angular momentum of the body about that point. Which of the following statements are correct? (a) dL/dt 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.
- 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 velocity v0, the ring moves without bouncing?