Which of the following will not be affected if the radius of the sphere is increased while keeping mass constant?

Three objects, A: (a solid sphere), B: (a thin circular disk) and C: (a circular ring), each have the same mass \(\mathrm{M}\) and radius \(\mathrm{R}.\) They all spin with the same angular speed about their own symmetry axes. The amount of work (\(\mathrm{W}\)) required to bring them to rest, would satisfy the relation:

The moment of the force, \(\overset{\rightarrow}{F} = 4 \hat{i} + 5 \hat{j} - 6 \hat{k}\) at point (\(2,\) \(0,\) \(-3\)) about the point (\(2,\) \(-2,\) \(-2\)) is given by:
1. \(- 8 \hat{i} - 4 \hat{j} - 7 \hat{k}\)

2. \(- 4 \hat{i} - \hat{j} - 8 \hat{k}\)

3. \(- 7 \hat{i} - 8 \hat{j} - 4 \hat{k}\)

4. \(- 7 \hat{i} - 4 \hat{j} - 8 \hat{k}\)

A solid sphere is rotating freely about its axis of symmetry in free space. The radius of the sphere is increased keeping its mass the same. Which of the following physical quantities would remain constant for the sphere?

A rope is wrapped around a hollow cylinder with a mass of \(3\) kg and a radius of \(40\) cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of \(30\) N?
1. \(0.25 ~\mathrm{rad} / \mathrm{s}^2\)
2. \(25 ~\mathrm{rad} / \mathrm{s}^2\)
3. \(5 ~\mathrm{m} / \mathrm{s}^2\)
4. \(25 ~\mathrm{m} / \mathrm{s}^2\)

Two discs of the same moment of inertia are rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ${\omega }_1$and ${\omega }_2$. They are brought into contact face to face with their axis of rotation coinciding. The expression for loss of energy during this process is:-

1. $\frac{1}{4}I{\left({\omega }_1-{\omega }_2\right)}^2$

2. $I{\left({\omega }_1-{\omega }_2\right)}^2$

3. $\frac{1}{8}I{\left({\omega }_1-{\omega }_2\right)}^2$

4. $\frac{1}{2}I{\left({\omega }_1-{\omega }_2\right)}^2$

Which of the following statements are correct? 

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. $L_A=\frac{L_B}{2}$
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 soild cyclinder 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 $\left(E_{sphere}/E_{cylinder}\right)$ $$ will be 

(a) 2:3

(b) 1:5

(c) 1:4

(d) 3:1

A bullet of mass \(10\) g moving horizontal with a velocity of \(400\) m/s strikes a wood block of mass \(2\) kg which is suspended by light inextensible string of length \(5\) m. As a result, the centre of gravity of the block is found to rise a vertical distance of \(10\) cm. The speed of the bullet after it emerges horizontally from the block will be: