For which of the following does the centre of mass lie outside the body?

1. A pencil             2. A shotput              3. A dice                4. A bangle

Which of the following points is the likely position of the center of mass of the system shown in the figure?

1. A
2. B

3. C

4. D

A particle of mass m is moving in yz-plane with a uniform velocity v with its trajectory running parallel to the +ve y-axis and intersecting z-axis at z a in the figure. The change in its angular momentum about the origin as it bounces elastically from a wall at y = constant is

1. mvae^x
2. 2mvae^x
3. ymvae^x
4. 2ymvae^x

When a disc rotates with uniform angular velocity, which of the following is not true?

1. The sense of rotation remains the same
2. The orientation of the axis of rotation remains the same
3. The speed of rotation is non-zero and remains the same
4. The angular acceleration is non-zero and remains the same

A uniform square plate has a small piece Q of an irregular shape removed and glued to the center of the plate leaving a hole behind in the figure. The moment of inertia about the z-axis is then,

1. increased
2. decreased
3. the same
4. changed in manner

A uniform square plate has a small piece Q of an irregular shape removed and glued to the center of the plate leaving a hole behind in the figure. The CM of the plate is now in the following quadrant of the x-y plane. 

1. I

2. II

3. III

4. IV

The density of a non-uniform rod of length 1m is given by $\mathrm{\rho }\left(\mathrm{x}\right)=\mathrm{a}(1+{\mathrm{bx}}^2)$ where, a, and b are constants and $0\le x\le 1$. The centre of mass of the rod will be at

1. \(\frac{3(2+b)}{4(3+b)}\)
2. \(\frac{4(2+b)}{3(3+b)}\)
3. \(\frac{3(3+b)}{4(2+b)}\)
4. \(\frac{4(3+b)}{3(2+b)}\)

A merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with the angular speed $\mathrm{\omega }$. A person of mass M is standing on its edge. At one instant, the person jumps off the round such that the final speed of the person is zero. What is the final angular velocity of the merry-go-round?

$\begin{array}{l}\text{1. }2\mathrm{\omega }\end{array}$
2. ω 
3. ω/2
4. 0

The figure below shows two identical particles 1 and 2, each of mass m, moving in opposite directions with the same speed v along parallel lines. At a particular instant, r₁ and r₂ are their respective position vectors drawn from point A which is in the plane of the parallel lines.

(a) Angular momentum $I_1$ Of particle 1 about A is $\mathrm{I}=\mathrm{mv}\left({\mathrm{d}}_1\right)\odot$
(b) Angular momentum $I_2$ of particle 2 about A is ${\mathrm{I}}_2={\mathrm{mvr}}_2\odot$
(c) Total angular momentum of the system about A is $\mathrm{I}=\mathrm{mv}({\mathrm{r}}_1+{\mathrm{r}}_2)\odot$
(d) Total angular momentum of the system about A is $\mathrm{I}=\mathrm{mv}({\mathrm{d}}_2-{\mathrm{d}}_1)\otimes$

Choose the correct option.

1. (a, c)

2. (a, d)

3. (b, d)

4. (b, c)

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

(a) The forces may be acting radially from a point on the axis
(b) The forces may be acting on the axis of rotation
(c) The forces may be acting parallel to the axis of rotation
(d) The torque caused by some forces may be equal and opposite to that caused by other forces

Choose the correct option.

1. (a, c, d)

2. (a, b, d)

3. (a, b, c, d)

4. (b, c, d)