The moment of inertia of a thin uniform rod of mass M and length L about an axis passing through its mid-point and perpendicular to its length is I₀. Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is:

1.  ${\mathrm{I}}_0+{\mathrm{ML}}^2/4$

2.  ${\mathrm{I}}_0+2{\mathrm{ML}}^2$

3.  ${\mathrm{I}}_0+{\mathrm{ML}}^2$

4.  ${\mathrm{I}}_0+{\mathrm{ML}}^2/2$

A circular disk of a moment of inertia I_t is rotating in a horizontal plane, about its symmetric axis, with a constant angular speed ${\mathrm{\omega }}_{\mathrm{i}}$. Another disk of a moment of inertia I_b is dropped coaxially onto the rotating disk. Initially, the second disk has zero angular speed. Eventually, both the disks rotate with a constant angular speed ${\mathrm{\omega }}_{\mathrm{f}}$. The energy lost by the initially rotating disc due to friction is:

1. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}^2}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^2$

2. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{t}}^2}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^2$

3. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}-{\mathrm{I}}_{\mathrm{t}}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^2$

4. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}{\mathrm{I}}_{\mathrm{t}}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^2$

Two particles that are initially at rest, move towards each other under the action of their mutual attraction. If their speeds are v and 2v at any instant, then the speed of the centre of mass of the system will be:

1. 2v

2. 0

3. 1.5v

4. v

A man of 50 kg mass is standing in a gravity-free space at a height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed of 2 ms^-1. When the stone reaches the floor, the distance of the man above the floor will be:

1. 9.9 m

2. 10.1 m

3. 10 m

4. 20 m

If $\overrightarrow{\mathrm{F}}$ is the force acting on a particle having position vector $\overrightarrow{r}$ and $\overrightarrow{\tau }$ be the torque of this force about the origin, then:

1. $\overrightarrow{r}$. $\overrightarrow{\tau }$≠ 0 and $\overrightarrow{F}$. $\overrightarrow{\tau }$ = 0

2. $\overrightarrow{r}$. $\overrightarrow{\tau }$> 0 and $\overrightarrow{F}$. $\overrightarrow{\tau }$< 0

3. $\overrightarrow{r}$. $\overrightarrow{\tau }$ = 0 and $\overrightarrow{F}$. $\overrightarrow{\tau }$ = 0

4. $\overrightarrow{r}$. $\overrightarrow{\tau }$= 0 and $\overrightarrow{F}$. $\overrightarrow{\tau }$≠ 0

A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m are attached gently to the opposite ends of the diameter of the ring, the ring will then rotate with an angular velocity:

1. $\frac{\omega (M-2m)}{M+2m}$

2. $\frac{\omega M}{M+2m}$

3. $\frac{\omega (M+2m)}{M}$

4. $\frac{\omega M}{M+m}$

Two bodies of mass 1 kg and 3 kg have position vectors $\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}} \mathrm{and} -3\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ respectively. The centre of mass of this system has a position vector:
1. -2$\hat{i}$+2$\hat{k}$
2. -2$\hat{i}$-$\hat{j}$+$\hat{k}$
3. 2$\hat{i}$- $\hat{j}$- 2$\hat{k}$
4. -$\hat{i}$+$\hat{j}$+$\hat{k}$

The ratio of the radii of gyration of a circular disc to that of a circular ring, each of the same mass and radius, around their respective axes is:

1.  $\sqrt{3} : \sqrt{2}$

2.  $1 : \sqrt{2}$

3.  $\sqrt{2} : 1$

4.  $\sqrt{2} : \sqrt{3}$

A thin rod of length L and mass M is bent at its midpoint into two halves so that the angle between them is 90^o. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is:

1.  $\frac{{\mathrm{ML}}^2}{24}$

2.  $\frac{{\mathrm{ML}}^2}{12}$

3.  $\frac{{\mathrm{ML}}^2}{6}$

4.  $\frac{\sqrt{2}{\mathrm{ML}}^2}{24}$