**Rotational Mechanics**

- This video introduces the concept of rotational motion and underlines its difference with translational motion and relative parameters.
- This video explains the different parameters of the rotational motion like angular displacement, angular velocity and angular acceleration. It also derives the relation between the parameters in equivalence to the equations of motion in translational motion.
- Find the force exerted by the hinge for the case where (a) a rod of length is tied to a mass-less string at its centre of mass and is rotating in a circle where the rod remain always perpendicular to the string as shown in figure (b) a uniform rod of length l is rotating about an axis passing through its centre of mass and perpendicular to the rod (c) A tube of length l is filled completely with an incompressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends. Find the force exerted by the liquid at the other end. Take the angular velocity to be ω in all cases. Ignore friction.
- This video describes the rotational kinetic energy of a rotating body in terms of moment of inertia which depends upon the axis of rotation.
- Find the moment of inertia of a particle of mass M situated at a distance R from the axis of rotation. How does the moment of inertia change if

a) mass is doubled

b) radius is doubled

c) mass is re-distribued keeping the radius the same - Tell whether the following statement is true/false. As we can consider the mass of any object to be concentrated at its centre of mass, therefore we can write the moment of inertia of any object as Mrcom2.
- This video describes the moment of inertia in detail and its dependence on the axis of rotation. It also explains the parallel axis theorem for moment of inertia.
- Tell whether the following statement is true/false. If two different axes are at same distance from the centre of mass of a rigid body, then moment of inertia of the given rigid body about both the axes will always be the same.
- The figure shows a uniform rod lying along the x-axis. The locus of all the points lying on the x-y plane, about which the moment of inertia of the rod is same as that about O, will be ..?
- Find the moment of inertia of a thin uniform rod of mass M and length L about an axis perpendicular to the rod and passing through one of its ends. Given that the moment of inertia of the rod about a perpendicular axis through its Centre of Mass is = (1/12) ML2
- This video explains the perpendicular axis theorem for finding moment of inertia when it is given for two perpendicular axis of rotation.
- Find moment of inertia of uniform rectangular plate about an axis perpendicular to the plane of plate and (a) passing through its centre (b) passing through its corner.
- This video explains the concept of torque which is responsible for bringing rotational motion in an object and method to determine its direction by right hand thumb rule.
- Discuss the possible cases of line of action of force with respect to the axis of rotation.
- This video discusses the relation between the force, torque, centre of mass and centre of gravity.
- Let F be a force acting on a particle having position vector r. Let t be the torque of this force about the origin. Which of the following is correct.
- This video explains the laws of rotational motion in terms of torque, moment of inertia and angular acceleration in equivalence with the laws of translational motion.
- Tell whether the following statements are true/false

(a) If the net external force on a body is zero, then its angular acceleration is zero. (b) If there is no external torque on body about its centre of mass, then the velocity of the centre of mass remains constant. - A thread is wound over a ring, disc, and sphere of same mass M and radius R and is free to rotate about the axis passing through their Centre of Mass. The free end of the thread is pulled with a constant force F as shown in figure. (a) Find the average acceleration in all cases. (the thread does not slip) (b) Angular velocity of each object when a length l of the thread is uncovered (ignore friction at the axle)
- Figure shows a uniform disk, with mass M and Radius R, moved on fixed horizontal axle. A block with mass m hangs from mass-less cord that is wrapped around the rim of disk. Find the (a) acceleration of the falling block, (b) angular acceleration of the disk, and (c) tension in the cord. (Assuming the cord does not slip and there is no friction at the axle)
- This video describes the work done in a rotational motion as the change in rotational kinetic energy and the power in term of torque and angular velocity.
- A uniform rod of mass m and length l is hinged about one of its ends and is kept in horizontal position as shown in the figure. Just after the rod is released, find the following. (a) The angular acceleration just after release. (b) Acceleration of Centre of Mass of the rod. (c) Acceleration of point B. (d) normal reaction from the hinge. (e) Angular velocity of the rod when it becomes vertical.
- This video explains the concept of angular momentum and the method to derive its formula and direction.
- Find the angular momentum of the particle in the situations as shown in figure with respect to point O and point B.
- This video explains the method to derive the rate of change of angular momentum with time which comes out to be equal to net torque on a body.
- An object of mass m is falling freely under the influence of gravity. Relate the torque and angular momentum of the object with respect to point O as shown in the figure.
- This video describes the method to obtain angular momentum of a rigid body and system of particles.
- This video explains the conservation of angular momentum in absence of any external torque and formula for impulse in case of change in angular momentum in a very short time.
- A frictionless rod is rotating in horizontal plane about an axis passing through one of its end with angular velocity ω0. A ball of mass m, free to slide on the rod, is initially held close to the axis of rotation and then released. Find (a) the angular velocity of the rod+ball system when ball reaches the other end of the rod. (b) difference in energy and explain this difference in energy. (c) speed with which the ball strikes the other end.
- Disk 1 rotating about a smooth vertical axis with the angular velocity w1 is kept on a stationary Disk 2 as shown in figure. Find the final angular velocity of the disks, given that the moments of inertia of the disks relative to the rotation axis are equal to I1 and I2, respectively. The contact surface of the disks is rough.
- Figure shows a step pulley P1 and P2 connected by a cross belt. If the angular acceleration of pulley is given, find the time required for A to travel a distance l from rest. Also, Find the distance moved by B in the same time.
- A thin uniform disc has mass M and radius R. A circular hole of radius R/3 is made in the disc as shown in the figure. The moment of inertia of the remaining portion of disc about an axis passing through centre O and perpendicular to the plane of the disc is
- I1 , I2 , I3 and I4 are respectively the moment of inertia of a thin square plate of uniform thickness about axes 1, 2, 3 and 4 which are in the plane of the plate as shown in figure. The moment of inertia of the plate about an axis passing through the centre O and perpendicular to the plane of the plate will be.....?
- Moment of inertia of the semicircular ring of mass M and radius R about an axis as shown in figure is......?
- Consider the system shown in the figure with blocks of masses M1, M2, and uniform pulley with moment of inertia I and radius R. The string is light and inextensible and does not slip on the pulley. The pulley axis is frictionless. Find the acceleration of the blocks.
- A pulley of radius R and moment of inertia I about its axis is fixed at the top of a frictionless plane inclined at an angle as shown in figure. A string is wrapped round the pulley and its free end supports a block of mass M which can slide on the incline. Initially, the pulley is rotating at a speed ω in a direction such that the block slides up the plane. How far will the block move before stopping?
- A uniform sphere of given mass and radius can rotate about a vertical axis on frictionless bearings. A mass-less cord passes around the equator of the sphere, over a pulley and is attached to a small object. There is no friction on the pulley's axle; the cord does not slip on the pulley and sphere. What is the speed of the object when it has fallen through a height h after being released from rest?
- Consider the system shown in the figure with blocks of masses m1, m2, and uniform pulley with moment of inertia I and inner radius r and outer radius R. The string is light and inextensible and does not slip on the pulley. The pulley axis is frictionless. Find the velocity of block with mass m2 just before it strikes with the base of clamp.
- A uniform rod is hinged at one of its ends in the horizontal position as shown in figure. The other end is connected to a block through a mass-less string passing over a uniform disc with moment of inertia equal to I. Acceleration of the block just after it is released from this position is...?
- A man of mass m stands on a horizontal platform in the shape of a disk of mass M and radius R, pivoted on a vertical axis through its centre about which it can freely rotate. The man starts to move around the centre of the disk in a circle of radius with a velocity v relative to the disk. Calculate the angular velocity of the disk.
- A thin uniform rod of length l and mass M is rotating horizontally in counter-clockwise direction with angular velocity ω about an axis through its centre. A particle of mass m hits the rod with velocity v0 and sticks to the rod. The particle's path is perpendicular to the rod at the instant of the hit, and at a distance d from the rod's centre. Answer the following.

(a) At what value of d are rod and particle stationary after the hit?

(b) In which direction do rod and particle rotate if d is greater than this value? - A uniform disc of mass M and radius R is hanging from a rigid support and is free to rotate about horizontal axis passing through its centre in vertical plane as shown in figure. An insect of mass m strikes the disc at a point at horizontal diameter with a velocity at an angle so that the disc completes the vertical circular motion. Find the velocity of the insect with which it strikes the disc.
- A horizontal, homogeneous cylinder of mass M and radius pivoted about its axis of symmetry. As shown in figure, a string is wrapped several times around the cylinder and tied to a body of mass m resting on a support positioned so that the string has no slack. The body of mass m is carefully lifted vertically to a distance h, and then released. Find the velocity of the block and the angular velocity of the cylinder just after the string become taut.
- A vertically oriented uniform rod of mass M and length l can rotate about its upper end. A horizontally flying bullet of mass m strikes the lower end of the rod and gets stuck in it; as a result, the rod swings through a given angle. Find the following (a) the velocity of the flying bullet (b) the momentum change in the system'bullet-rod' during the impact ; and the reason for that change. (c) at what distance x from the upper end of the rod the bullet must strike for the momentum of the system "bullet-rod" to remain constant during the impact?
- A uniform rod of mass M and length l is lying on a frictionless horizontal plane and is hinged about one of its end. A particle of mass m strikes the rod at a distance of 3l/4 from the hinge. If the co-efficient of restitution of collision is e, find the velocity of particle and angular velocity of the rod just after the collision.
- One side of a spring of initial un-stretched length lo lying on a friction less surface is fixed. The other end is fastened to a small puck of mass m. The puck is given initial velocity v0 in direction perpendicular to the spring. In the course of motion, the maximum elongation of the spring l is known. What is force constant of the spring?
- Figure shown a mass m placed on a frictionless horizontal table and attached to a string passing through a small hole in the surface. Initially, the mass moves in a circle of radius ro with a speed vo and the free end of the string is held by a person. The person pulls on the string slowly to decrease the radius of the circle to r. (a) Find the tension in the string when the mass moves in the circle of radius r. (b) Calculate the change in the kinetic energy of the mass. (c) Verify that the work done on system is equal to the change in its Kinetic Energy.
- A uniform disc of mass M and radius R is hinged about a point along its edge and is free to rotate in the vertical plane. Initially the disc is held such that the hinge and the Centre of Mass of the disc lie along a horizontal line as shown in the figure. After the disc is released, find the following as a function of angle θ (a) angular velocity of the disc (b) angular acceleration of the disc (c) Force exerted by the hinge (d) Force exerted by the half of the disc closer to hinge on the other half.
- A particle of mass m is projected with a speed u at an angle θ to the horizontal at time t = 0. Find its angular momentum about the point of projection O at time t, vectorially. Assume the horizontal and vertical lines through O as X and Y axes.
- Consider a uniform rod of mass is M and length L. What is the rotational inertia of the rod about the following axes? (a) the perpendicular axis through the centre. (b) an axis passing through centre at an angle α with the rod.
- Find the moment of inertia of a thin uniform rectangular plate of mass M and given dimension, about an axis parallel to smaller side of the plate and passing through its centre of mass.
- Find the moment of inertia of a uniform circular plate about an axis passing through its centre of mass and (a) is perpendicular to the plane of the plate (b) lies in the plane of plate.
- Find the moment of inertia of (a) a uniform hollow sphere about a diameter. (b) a uniform solid sphere about a diameter.
- Point masses M1 and M2 are placed at the opposite ends of rigid rod of length L and negligible mass. The rod is rotating about an axis perpendicular to it. Find the position on this rod through which the axis should pass so that the Kinetic energy of the system is minimum for a given angular velocity.
- A uniform cylinder of radius R and mass M can rotate freely about a fixed horizontal axis. A thin cord of length L and mass m is wound on the cylinder in a single layer. Find the angular acceleration of the cylinder as a function of the length x of the hanging part of the cord. The wound part of the cord is supposed to have its centre of gravity on the cylinder axis as shown figure.
- A uniform disc of radius R is snipped to a given angular velocity and then carefully placed on a horizontal surface. How long will the disc be rotating on a rough surface with known coefficient of friction. The pressure exerted by the disc on the surface can be regarded as uniform.