A solid cylinder of mass M and radius R rolls down an inclined plane of height h without slipping. The speed of its centre of mass when it reaches the bottom is

A ball is travelling with uniform translatory motion. This means that:

A metre scale is moving with uniform velocity. This implies:

During inelastic collision between two bodies, which of the following quantities always remain conserved?

Internal forces can change:

A pulley is hinged at the centre and a massless thread is wrapped around it. The thread is pulled with a constant force F starting from rest. As the time increases:

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

Q. 1 For which of the following does the centre of mass lie outside the body?

(a) A pencil             (b) A shotput              (c) A dice                (d) A bangle

Q. 2 Which of the following points is the likely position of the center of mass of the system shown in the figure?

(a) A

(b) B

(c) C

(d) D

Q. 3 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

$\begin{array}{llll}\text{ (a) }\mathrm{mva}{\hat{\mathrm{e}}}_{\mathrm{x}}& \text{ (b) }2\text{ mva }{\hat{\mathrm{e}}}_{\mathrm{x}}& \text{ (c) ymv }{\hat{\mathrm{e}}}_{\mathrm{x}}& \text{ (d) }2\mathrm{ymv}{\hat{\mathrm{e}}}_{\mathrm{x}}\end{array}$

Q. 4 When a disc rotates with uniform angular velocity, which of the following is not true?

(a) The sense of rotation remains the same
(b) The orientation of the axis of rotation remains the same
(c) The speed of rotation is non-zero and remains the same
(d) The angular acceleration is non-zero and remains the same

Q. 5 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,

(a) increased
(b) decreased
(c) the same
(d) changed in manner

In problem 5, the CM of the plate is now in the following quadrant of the x-y plane. 

(a) I

(b) II

(c) III

(d) IV

Q. 7 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

$\begin{array}{llll}\text{ (a) }\frac{3(2+\mathrm{b})}{4(3+\mathrm{b})}& \text{ (b) }\frac{4(2+\mathrm{b})}{3(3+\mathrm{b})}& \text{ (c) }\frac{3(3+\mathrm{b})}{4(2+\mathrm{b})}& \text{ (d) }\frac{4(3+\mathrm{b})}{3(2+\mathrm{b})}\end{array}$

Q. 8 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?

The angular speed of the wheel of a vehicle is increased from 360 rpm to 1200 rpm in 14 seconds. Its angular acceleration will be:

Three identical spheres, each of mass M, are placed at the corners of a right-angle triangle with mutually perpendicular sides equal to 2 m (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin, find the position vector of the centre of mass. 

Three particles of masses 100g, 150g, and 200g respectively are placed at the vertices of an equilateral triangle of a side 0.5 m long. What is the position of the centre of mass of three particles?

Position of centre of mass of a triangular lamina as shown in the figure is?

Q. 10 Figure 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. Choose the correct options.

(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 $\mathrm{I}=\mathrm{mv}({\mathrm{d}}_2-{\mathrm{d}}_1)\otimes$

Q. 12 Figure shows a lamina in XY-plane. Two axes z and z' pass perpendicular to its plane. A force 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'-axis than the z-axis.)

(a) Torque $\overrightarrow{\mathrm{\tau }}$caused by $\overrightarrow{\mathrm{F}}$ about z-axis is along $-\hat{\mathrm{k}}$
(b) Torque $\overrightarrow{\mathrm{\tau }}$' caused by $\overrightarrow{\mathrm{F}}$ about z'-axis is along $-\hat{\mathrm{k}}$
(c) Torque caused by $\overrightarrow{\mathrm{F}}$ about the z-axis is greater in magnitude than that about the z'-axis
(d) Total torque is given be ${\overrightarrow{\mathrm{\tau }}}_{\mathrm{net}}$= $\overrightarrow{\mathrm{\tau }}$+$\overrightarrow{\mathrm{\tau }}$'

A cricket ball of mass 150 g moving with a speed of 126 km/h hits at the middle of the bat, held firmly at its position by the batsman. The ball moves straight back to the bowler after hitting the bat. Assuming that collision between ball and bat is completely elastic and the two remains in contact for 0.001s, the force that the batsman had to apply to hold the bat firmly at its place would be

During inelastic collision between two bodies, which of the following
quantities always remain conserved?

Q. 9 Two identical ball bearings in contact with each other and resting on a frictionless table is hit head-on by another ball bearing of the same time a  moving initially with a speed v as shown in the figure.

If the collision is elastic, which of the following (figure) is a possible result after collision?

Two blocks ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ having equal mass are free to move on a horizontal frictionless surface.  is attached to a massless spring as shown in the figure. Initially ${\mathrm{M}}_2$ is at rest and ${\mathrm{M}}_1$ is moving toward ${\mathrm{M}}_2$ with speed v and collides head-on with ${\mathrm{M}}_2$.

Q. 16 The centre of mass of an extended body on the surface of the earth and its centre of gravity:

(a) are always at the same point for any size of the body.

(b) are always at the same point only for spherical bodies.

(c) can never be at the same point.

(d) is close to each other for objects, say of sizes less than 100 m.

(e) both can change if the object is taken deep inside the earth.

Two billiard balls A and B, each of mass 50g and moving in opposite directions with a speed of 5 m/s each, collide and rebound at the same speed. If the collision lasts for ${10}^{-3}$ s, which of the following statements are true?

(a) The impulse imparted to each ball is 0.25 $kg-m{s}^{-1}$ and the force on each ball is 250 N

(b) The impulse imparted to each ball is 0.25 $kg-m{s}^{-1}$ and the force exerted on each ball is 25 x ${10}^{-5}$

(c) The impulse imparted to each ball is 0.5 N-s

(d) The impulse and the force on each ball are equal in magnitude and opposite in directions.

What is the position of centre of mass of a uniform L-shaped lamina (a thin flat plate) with dimensions as shown? The mass of the lamina is 3 kg.

If the force $7\hat{\mathrm{i}}+3\hat{\mathrm{j}}-5\hat{\mathrm{k}}$ acts on a particle whose position vector is $\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$, the torque of the force about the origin is?

The angular momentum about any point of a single particle moving with constant velocity: