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A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere?
1. Angular velocity 
2. Moment of inertia
3. Angular momentum 
4. Rotational kinetic energy

A solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy (K_t) as well as rotational kinetic energy (K_r) simultaneously. The ratio K_t : (K_t + K_r) for the sphere is 
1. 7 : 10 
2. 5 : 7
3. 2 : 5 
4. 10 : 7

Three objects, A : (a solid sphere), B : (a thin circular disk) and C : (a circular ring), each have the same mass M and radius R. They all spin with the same angular speed ω about their own symmetry axes. The
amounts of work (W) required to bring them to rest, would satisfy the relation
1. W_C > W_B > W_A 
2. W_A > W_B > W_C
3. W_A > W_C > W_B 
4. W_B > W_A > W_C

The moment of the force, F =$4\hat{\mathrm{i}}+5\hat{\mathrm{j}}-6\hat{\mathrm{k}}$ at (2, 0, –3), about the point (2, –2, –2), is given by 

1. $-8\hat{\mathrm{i}}-4\hat{\mathrm{j}}-7\hat{\mathrm{k}}$
2. $-4\hat{\mathrm{i}}-\hat{\mathrm{j}}-8\hat{\mathrm{k}}$
3. $-7\hat{\mathrm{i}}-4\hat{\mathrm{j}}-8\hat{\mathrm{k}}$
4. $-7\hat{\mathrm{i}}-8\hat{\mathrm{j}}-4\hat{\mathrm{k}}$

A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? 
1. 25 m/s² 
2. 0.25 rad/s² 
3. 25 rad/s² 
4. 5 m/s²

Which of the following statements are correct?
(a) Centre of mass of a body always coincides with the centre of gravity of the body.
(b) Centre of mass of a body is the point at which the total gravitational torque on the body is zero
(c) A couple on a body produce both translational and rotational motion in a body.
(d) Mechanical advantage greater than one means that small effort can be used to lift a large load.
1. (b) and (d)                2. (a) and (b) 
3. (b) and (c)                4. (c) and (d)

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ω₁ and ω₂. They are brought into contact face to face coinciding
the axis of rotation. The expression for loss of energy during this process is
1. $\frac{1}{2}l{\left({\omega }_1+{\omega }_2\right)}^2$
2. $\frac{1}{4}l{\left({\omega }_1\mathit{-}{\omega }_2\right)}^2$
3. $l{\left({\omega }_1\mathit{-}{\omega }_2\right)}^2$
4. $\frac{l}{8}{\left({\omega }_1\mathit{-}{\omega }_2\right)}^2$

Two rotating bodies A and B of masses m and 2m with moments of inertia I_A and I_B (I_B > I_A) have equal kinetic energy of rotation. If L_A and L_B be their angular momenta respectively, then 

1. ${\mathrm{L}}_{\mathrm{A}}=\frac{{\mathrm{L}}_{\mathrm{B}}}{2}$
2. L_A = 2L_B 
3. L_B > L_A 
4. L_A > L_B

A solid sphere of mass m and radius R is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation (E_sphere / E_cylinder) will be 
1. 2 : 3 
2. 1 : 5
3. 1 : 4 
4. 3 : 1

A light rod of length l has two masses m1 and m2 attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is
1. $\frac{{\mathrm{m}}_1{\mathrm{m}}_2}{{\mathrm{m}}_1+{\mathrm{m}}_2}{l}^2$
2. $\frac{{\mathrm{m}}_1+{\mathrm{m}}_2}{{\mathrm{m}}_1{\mathrm{m}}_2}{l}^2$
3. $\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)l^2$
4. $\sqrt{{\mathrm{m}}_1{\mathrm{m}}_2} l^2$

From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through
the centre?
1. $\frac{9{\mathrm{MR}}^2}{32}$
2. $\frac{15{\mathrm{MR}}^2}{32}$
3. $\frac{13{\mathrm{MR}}^2}{32}$
4. $\frac{11{\mathrm{MR}}^2}{32}$

A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first? 
1. Depends on their masses 
2. Disk
3. Sphere 
4. Both reach at the same time

An automobile moves on a road with a speed of 54 km h^–1. The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 kg m². If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is 
1. 2.86 kg ${\mathrm{m}}^2{\mathrm{s}}^{-2}$ 
2. 6.66 kg ${\mathrm{m}}^2{\mathrm{s}}^{-2}$
3. 8.58 kg ${\mathrm{m}}^2{\mathrm{s}}^{-2}$ 
4. 10.86 kg ${\mathrm{m}}^2{\mathrm{s}}^{-2}$

Point masses m₁ and m₂ are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which
the axis should pass so that the work required to set the rod rotating with angular velocity ω₀ is minimum, is given by

1. $\mathrm{x}=\frac{{\mathrm{m}}_2\mathrm{L}}{{\mathrm{m}}_1+{\mathrm{m}}_2}$
2. $\mathrm{x}=\frac{{\mathrm{m}}_1\mathrm{L}}{{\mathrm{m}}_1+{\mathrm{m}}_2}$
3. $\mathrm{x}=\frac{{\mathrm{m}}_1\mathrm{L}}{{\mathrm{m}}_2}$
4. $\mathrm{x}=\frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}$

A force F = $\mathrm{\alpha }\hat{\mathrm{i}}+3\hat{\mathrm{j}}+6\hat{\mathrm{k}}$  is acting at a point $\overrightarrow{\mathrm{r}}=2\hat{i}-6\hat{j}-12\hat{k}$.  The value of $\mathrm{\alpha }$ for which angular momentum about origin is conserved is 
1. 1 
2. –1
3. 2 
4. Zero

A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position.The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The
normal reaction on A is
1. $\frac{W\left(d-x\right)}{d}$
2. $\frac{Wx}{d}$
3. $\frac{Wd}{x}$
4. $\frac{W\left(d-x\right)}{x}$

A mass m moves in a circle on a smooth horizontal plane with velocity v0 at a radius R₀. The mass is attached to a string which passes through a smooth hole in the plane as shown.

The tension in the string is increased gradually and finally m moves in a circle of radius $\frac{{\mathrm{R}}_0}{2}$. The final value of the kinetic energy is
1. $\frac{1}{2}m{v}_0^2$
2. ${\mathrm{mv}}_0^2$
3. $\frac{1}{4}m{v}_0^2$
4. $2{\mathrm{mv}}_0^2$

Three identical spherical shells, each of mass \(m\) and radius \(r\) are placed as shown in figure. Consider an axis \(XX'\) which is touching to two shells and passing through diameter to third shell. Moment of inertia of the system
consisting of these three spherical shells about \(XX'\) axis is:
 
1. \(4mr^2\)
2. \(\dfrac{11}{5}mr^2\)
3. \(3mr^2\)
4. \(\dfrac{16}{5}mr^2\)

Two spherical bodies of mass M and 5M and radii R and 2R are released in free space with initial separation between their centres equal to 12 R. If they attract each other due to gravitational force only, then the distance
covered by the smaller body before collision is 
1. 1.5R 
2. 2.5R 
3. 4.5R 
4. 7.5R

A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required
to produce an angular acceleration of 2 rev/s² is 
1. 25 N                  2. 50 N 
3. 78.5 N               4. 157 N

The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle θ without slipping and slipping down the incline without rolling is 
1. 5 : 7                2. 2 : 3 
3. 2 : 5                4. 7 : 5

A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches upto a maximum height of $\frac{3{\mathrm{v}}^2}{4\mathrm{g}}$ with respect to the initial position. The object is: 
1. Solid sphere 
2. Hollow sphere 
3. Disc 
4. Ring

A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is:

1. $\frac{\mathrm{g}}{\mathrm{L}}$
2. $\frac{2\mathrm{g}}{\mathrm{L}}$
3. $\frac{2\mathrm{g}}{3\mathrm{L}}$
4. $\frac{3\mathrm{g}}{2\mathrm{L}}$

ABC is an equilateral triangle with O as its centre $\overrightarrow{{\mathrm{F}}_1},\overrightarrow{F_2} \mathrm{and} \overrightarrow{F_3}$ represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then the magnitude of $\overrightarrow{F_3}$ is

1. $\frac{{\mathrm{F}}_1+{\mathrm{F}}_2}{2}$
2. $2\left({\mathrm{F}}_1+{\mathrm{F}}_2\right)$
3. ${\mathrm{F}}_1+{\mathrm{F}}_2$
4. ${\mathrm{F}}_1-{\mathrm{F}}_2$

When a mass is rotating in a plane about a fixed point, its angular momentum is directed along
1. The radius
2. The tangent to the orbit
3. A line perpendicular to the plane of rotation
4. The line making an angle of 45º to the plane of rotation