A small disk is rotating about an axis perpendicular to the plane of the disc and passing through its center $C$. The disk changed its angular speed from $7 \ rad/s$ to $12 \ rad/s$ in $3 \ s$. What is the angular displacement of the disk in this time period?

The below rod is in mechanical equilibrium.

$C$ represents the center of the rod, and it has a length of $20 \ cm$. Find the magnitude $F$ and the direction $θ$ for the force applied on the right edge.

Two hard spheres of mass $2 \ kg$ each are $1 \ m$ apart. When the distance between the two spheres is increased to $1.5 \ m$, the moment of inertia of the system of two spheres increases to:

(The axis of rotation passes through the midpoint of the line joining the two spheres.)

The angular displacement of a body is given by the formula,

$θ(t)=2t(1+t^3 )$

where $θ$ is in $rad$ and $t$ is in $s$. Which graph best represents the angular acceleration of the body?

A wheel of radius $4 \ cm$ is rolling on a surface without slipping with an angular velocity of $2 \ rad/s$, as shown in the figure below.

It is known that the net velocity of point A is the vector sum of the velocity of the center of mass and the (rotational) velocity of point A about the center of mass. What is the magnitude of the net velocity of point $A$?

(Hint: Velocity of Center of Mass for motion without slipping is $ω × r$.)

A body of moment of inertia $I$ undergoes rotational motion (starting from rest) due to constant torque $τ$ acting on it. If the angular displacement of the body is equal to $2πN$ (where $N$ is the number of revolutions of the body), find the average time period of the body.

Four particles of mass $m$ each are placed at the corners of a rectangle of length $a$ and width $b$. What is the moment of inertia of the system about an axis passing through one of the corners and perpendicular to the plane of the rectangle?

Find the angular acceleration of the pulley having a moment of inertia of $0.1 \ kgm^2$ in the figure below.

(Assume the rope does not slip on the pulley.)

A solid cylinder of mass $2.4 \ kg$ and radius $0.5 \ m$ is rolling on a horizontal surface without slipping. The angular velocity of the cylinder is $4 \ rad/s$. What is the total kinetic energy of the cylinder?

A non-uniform rod of mass density $ρ(x)=2Ax \ g/cm$ can rotate about point $Y$, as shown below.

What is the moment of inertia of the rod about point $Y$?

(Here, $x$ is the distance from the origin $O$, and $A$ is a positive constant.)

The moment of inertia of a solid sphere of mass $M$ and radius $R$ about an axis passing through its center is equal to $ \frac{2}{5} MR^2$. What would be the moment of inertia for an axis tangent to the sphere?

The diagram below shows the directions in which two particles of mass $m$ each move in an $xy$-plane.

The magnitude of the velocities of the particles, $X$ and $Y$, are $5 \ m/s$ and $6 \ m/s$, respectively. Determine the angular momentum of $X$ with respect to $Y$.

(Hint: What is the velocity of $X$ with respect to $Y$?)

Questions 13 and 14 are based on the below information:

A torque $τ(t) = 6t^2$ acts on a thin disc of moment of inertia $3 \ kgm^2$. The disc moves about an axis, passing through its center of mass and perpendicular to the plane of the disc.

What is the change in the angular momentum of the disc when it moves from time $t=2 \ s$ to $t=5 \ s$?

A torque $τ(t) = 6t^2$ acts on a thin disc of moment of inertia $3 \ kgm^2$. The disc moves about an axis, passing through its center of mass and perpendicular to the plane of the disc.

If the disc was momentarily at rest at time $t=2 \ s$, what is the angular displacement of the disc in the time interval $t=2 \ s$ to $t=5 \ s$?

Two solid discs are rotating about a common axis of rotation at speeds $8.5 \ rad/s$ and $-12.5 \ rad/s$. The masses of the discs are $4 \ kg$ and $3 \ kg$, respectively. The radius of both the discs is the same and is equal to $20 \ cm$.

If the two discs are combined to rotate together as shown in the diagram above, then what is the common angular velocity of the two discs?

(Hint: No external torque is needed to achieve this.)