What is Rotational Motion ? Are Newton?s Laws valid for rotational motion ? What is the difference between Rotation and Curvilinear Translation ?

Defining the rotational parameters like angular displacement, angular velocity and angular acceleration used to define a rotational motion. Is Angular Displacement a vector quantity ? Is Angular Velocity a vector quantity ? Equations for Rotational motion as a counterpart of Translational motion.

Find the force exerted by the hinge for the case where

a) a rod of length is tied to a massless string at its center of mass and is rotating as shown

b) a uniform rod of length l is rotating about an axis passing through its center 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 w in all cases. Ignore friction.

Kinetic Energy of a Rotating body. What is Moment of Inertia ? Does Moment of Inertia depend on 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

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

True / False

Parallel Axis Theorem helps us find the Moment of Inertia of a body wrt an axis parallel to an axis passing through the CoM. Is Moment of Inertia of a body minimum wrt any particular axis ?

If two different axes are at same distance from the center of mass of a rigid body, then moment of inertia of the given rigid body about both the axes will always be the same.

True / False

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, is:

a) an ellipse

b) a circle

c) parabola

d) a straight line

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

CoM is =

Perpendicular Axis Theorem relates the Moment of Inertia along x- y- and z-axis of Thin, Plate-like objects.

Find moment of inertia of uniform rectangular plate about an axis perpendicular to the plane of plate and

a) passing through its center

b) passing through its corner

Newton?s Laws give us a relation between the Net Force and Acceleration. But is the Point of Application of Force important ? Will the outcome be different if 2 equal and opposite forces are applied at the same point of an object or at different points ? Will Newton?s Laws be violated ? What is Torque ? Does it take into account the Force as well as the Point of Application of Force ?

Discuss the possible cases of line of action of force with respect to the axis of rotation.

Is Torque defined wrt an Axis of Rotation ? Or can it be defined wrt a point ? Can we write Net Torque = Center of Position x Net Force ?

Let F be a force acting on a particle having position vector r. Let t be the torque of this force about the origin.

Rotational Counterpart of Newton?s Laws of Motion.

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 center of mass, then the velocity of the center of mass remains constant.

A thread is wound over a ring, disc, and sphere of same mass M and radius R and free to rotate about the axis passing through their CoM.

The free end of the thread is pulled with a constant force F as shown.

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 massless 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.

(The cord does not slip, and there is no friction at the axle)

Rotational Counterpart of Work-Kinetic Energy Theorem

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 :

a) the angular acceleration just after release.

b) acceleration of CoM of the rod.

c) acceleration of point B.

d) normal reaction from the hinge.

e) angular velocity of the rod when it becomes vertical.

What is Angular Momentum ? Does Angular Momentum depend on the point of reference ?

Find the angular momentum of the particle in the given situations with respect to

(a) point O

(b) point B

Relation between Torque and Angular Momentum

An object of mass m is falling freely under the influence of gravity.

Relate the torque and angular momentum of the object wrt point O as shown in the figure.

Angular Momentum of Rigid Bodies rotating about an Axis

Conservation of Angular Momentum is the rotational counterpart of the Conservation of Linear Momentum. What is Angular Impulse and its relation with Angular Momentum ?

A frictionless rod is rotating in horizontal plane about an axis passing through one of its end with angular velocity wo. 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; explain the 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 stationery Disk 2. 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 are rough.

Figure shows a step pulley P1 and P2 connected by a cross belt. If the angular acceleration of pulley P2 be a2 rad/s2, 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 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. The moment of inertia of the plate about an axis passing through the center O and perpendicular to the plane of the plate is

(a) (I1 + I2)

(b) (I3 + I4)

(c) (I1 + I3)

(d) (I1 + I2 + I3 + I4)

Moment of inertia of the semicircular ring of mass M and radius R about an axis

as shown in figure

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 inclined plane of inclination q 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 w in a direction such that the block slides up the plane. How far will the block move before stopping?

A uniform sphere of mass Ms and radius R can rotate about a vertical axis on frictionless bearings. A massless cord passes around the equator of the sphere, over a pulley of mass Mp and radius r, and is attached to a small object of mass m. 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 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 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 massless 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 center about which it can freely rotate. The man starts to move around the center 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 counterclockwise direction with angular velocity w about an axis through its center. 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 center.

(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 2 v0 at an angle of 45? so that the disc completes the vertical circular motion.

Find v0.

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 an angle q. Find:

(a) the velocity of the flying bullet

(b) the momentum change in the system ?bullet-rod? during the impact ; what causes that change of momentum

(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 unstretched 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 = l0.

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 KE.

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 CoM 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 q

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 q 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,

(a) about the perpendicular axis through the center ?

(b) about an axis passing through center at an angle a with the rod ?

Find the moment of inertia of a thin uniform rectangular plate of mass M and dimensions a and b, about an axis parallel to side b 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 KE of the system is minimum for a given angular velocity wO.

A uniform cylinder of radius R and 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 center of gravity on the cylinder axis as shown figure.

A uniform disc of radius R is snipped to the angular velocity w0 and then carefully placed on a horizontal surface.

How long will the disc be rotating on a surface if the friction coefficient is equal

to m. The pressure exerted by the disc on the surface can be regarded as uniform.

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