Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of the 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

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?

• Angular velocity

• Moment of inertia

• Angular momentum

• Rotational Kinetic energy

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

• W_C > W_B> W_A

• W_A > W_B > W_C

• W_A>W_C>W_B

• W_B>W_A>W_C

A uniform rod of length l is free to rotate in a vertical plane about a fixed horizontal axis through B. The rod begins rotating from rest from its unstable equilibrium position. When, it has turned through an angle θ, its angular velocity ω is given by

• $\sqrt{\left(\frac{6\mathrm{g}}{\mathrm{I}}\right)} \sin \frac{\mathrm{\theta }}{2}$

• $\sqrt{\left(\frac{6\mathrm{g}}{\mathrm{I}}\right)} \cos \frac{\mathrm{\theta }}{2}$

• $\sqrt{\left(\frac{6\mathrm{g}}{\mathrm{I}}\right)} \sin \mathrm{\theta }$

• $\sqrt{\left(\frac{6\mathrm{g}}{\mathrm{l}}\right)} \cos \mathrm{\theta }$

ABC is the right-angled triangular plane of uniform thickness. The sides are such that AB>BC as shown in the figure. I₁, I₂, I₃ are moments of inertia about AB, BC and AC, respectively. Then which of the following relations is correct?

• I₁ = I₂ = I₃

• I₂ > I₁> I₃

• I₃<I₂<1₁

• I₃>I₁>I₂

An ice-berg of density 900 kgm^-3 is floating in the water of density 1000 kgm^-3. The percentage of the volume of ice-berg outside the water is

• 20%

• 35%

• 10%

• 11%

A solid cylinder is attached to a horizontal massless spring as shown in the figure. If the cylinder rolls without slipping, the time period of oscillation of the cylinder is

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{x}}{\mathrm{g}}}$

• $2\mathrm{\pi }\sqrt{\frac{2\mathrm{M}}{3\mathrm{K}}}$

• $2\mathrm{\pi }\sqrt{\frac{3\mathrm{M}}{8\mathrm{K}}}$

• $2\mathrm{\pi }\sqrt{\frac{3\mathrm{M}}{2\mathrm{K}}}$

48.

The moment of inertia of a body about a given axis 1.2 kgm²_• Initially the body is at rest. In order to produce a rotational kinetic energy of 1500 J and angular acceleration of 25 rad/s² must be applied about the axis for a duration of

• 10s

• 8s

• 2s

• 4s

Five particles of mass 2 kg are attached to the rim of a circular disc of radius 0.1 m and negligible mass. Moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane is

• 1 kg m²

• 0.1 kg m²

• 2 kg m²

• 0.2 kg m²

A particle is kept at rest at the top of a sphere of diameter 42 m. When disturbed slightly, it slides down. At what height h from the bottom, the particle will leave the sphere?

• 14 m

• 28 m

• 35 m

• 7 m