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

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

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

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

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

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

The angular momentum of a rotating body changes from A_o to 4A_o in 4 minutes. The torque acting on the body is:

• 3/4 A_o

• 4A_o

• 3A_o

• 3/2A_o

If a spherical ball rolls on a table without slipping the friction of its total energy associated with rotational energy is :

• 3/5

• 2/7

• 2/5

• 3/7

The equation of stationary wave along a stretched string is given by $\mathrm{y} = 5 \sin \frac{\mathrm{\pi x}}{3} \cos 40\mathrm{\pi t}$ where, x and y are in cm and t in second. The separation between two adjacent nodes is :

• 1.5 cm

• 3 cm

• 6 cm

• 4 cm

If the angle between the vectors $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}} \mathrm{is} \mathrm{\theta }$, the value of the product $\left(\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{A}}\right). \overrightarrow{\mathrm{A}}$ is equal to

• ${\mathrm{BA}}^2 \mathrm{cos\theta }$

• BA² sinθ

• BA² sinθ cosθ

• zero

The moment of inertia of a uniform circular disc of radius R and mass M about an axis passing from the edge of the disc and normal to the disc is

• $\frac{1}{2} {\mathrm{MR}}^2$

• MR²

• $\frac{7}{2} {\mathrm{MR}}^2$

• $\frac{3}{2} {\mathrm{MR}}^2$

A force of $-\mathrm{F} \hat{\mathrm{k}}$ acts on O, the origin of the coordinate systems. The torque about the point (1, -1 ) is

• $\mathrm{F} \left(\hat{\mathrm{i}} - \hat{\mathrm{j}}\right)$

• $- \mathrm{F} \left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• $\mathrm{F} \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} \right)$

• $-\mathrm{F} \left(\hat{\mathrm{i}} - \hat{\mathrm{j}}\right)$