Torque and Angular Momentum

A uniform sphere of mass $m$ and radius $R$ is placed on a rough horizontal surface Gigure). The sphere is struck horizontally at a height $h$ from the floor. Match the following:

A uniform disc of radius $R$, is resting on a table on its rim. The coefficient of friction between disc and table is $\mu$ (Figure).

Now the disc is pulled with a force $\overrightarrow{F}$ as shown in the figure. What is the maximum value of $F$ for which the dise rolls without slipping?

Two cylindrical hollow drums of radii $R$ and $2 R$, and of a common height $h$, are rotating with angular velocities $\omega$ (anti-clockwise) and $\omega$ (clockwise), respectively. Their axes, fixed are parallel and in a horizontal plane separated by $(3 R+\delta )$. They are now brought in contact $(\delta \to 0)$.

Identify forces and torques external to the system just after contact.

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to
the disc and passing through the centre), and rotating with angular speed $\omega 1$ and $\omega 2$ are brought into contact face to face with their axes of rotation coincident.

Find the angular speed of the two-disc system.

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to
the disc and passing through the centre), and rotating with angular speed $\omega 1$ and $\omega 2$ are brought into contact face to face with their axes of rotation coincident.

Does the law of conservation of angular momentum apply to the situation? why?

A door is hinged at one end and is free to rotate about a vertical axis . Does its weight cause any torque about this axis? Give reason for your answer.

The vector sum of a system of non-collinear forces acting on a rigid body is given to be non-zero. If the vector sum of all the torques due to the system of forces about a certain point is found to be zero, does this mean that it is necessarily zero about any arbitrary point?

Figure shows a lamina in $x-y$ plane. Two axes $z$ and $z^{\text{'}}$ pass perpendicular to its plane. A force $\overrightarrow{F}$ acts in the plane of lamina at point $P$ as shown. Which of the following are true? (The point $P$ is closer to $z^{\text{'}}$$-axis$ than the $z-axis$.)

The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?

A Merry-go-round, made of a ring-like platform of radius $\mathbf{R}$ and mass $\mathrm{M}$, is revolving with angular speed $\omega$. A person of mass $M$ is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round). The speed of the round afterwards is