A ball is moving in a circular path of radius 5 m. If tangential acceleration at any instant is 10 m/s and the net acceleration makes an angle 30° with the centripetal acceleration, then the instantaneous speed is

• $50\sqrt{3}m/s$

• $9.3m/s$

• $6.6\mathrm{m}/\mathrm{s}$

• $5.4\mathrm{m}/\mathrm{s}$

A spherical solid ball of mass 1 kg and radius 3 cm is rotating about an axis passing through its centre with an angular velocity of 50 rad/s. The kinetic energy of rotation is

• 4500J

• 90J

• 910J

• 0.45J

4. The moment of inertia of a thin spherical shell of mass M and radius R about a diameter is$\frac{2}{3}{\mathrm{MR}}^2$. Its radius of gyration K about a tangent will be

• $\sqrt{\frac{2}{3}}\mathrm{R}$

• $\frac{2}{3}\mathrm{R}$

• $\frac{5}{3}\mathrm{R}$

• $\sqrt{\frac{5}{3}}\mathrm{R}$

A body weighs 200 N at the surface of earth. If it be placed in an artificial satellite revolving at height where acceleration due to gravity is half of that at earth's surface. It will weigh

• 100N

• 200N

• 400N

• Zero

A wire suspended vertically from one of its ends is stretched by attaching a weight of 200 N to the lower end. The weight stretches the wire by 1 mm. Then the elastic energy stored in the wire is

• 20 J

• 0.1 J

• 0.2 J

• 10 J

Certain neutron stars are believed to be rotating at about 1 rev/s. If such a star has a radius of 20 km, the acceleration of an object on the equator of the star will be

• 20 $\times$10⁸ m/s²

• 8 $\times$10⁵ m/s²

• 120$\times$ 10⁵ m/s²

• 4 $\times$ 10⁸ m/s²

In planetary motion, the areal velocity of position vector of a planet depends on angular velocity ($\mathrm{\omega }$) and distance of the planet from the sun (r). If so the correct relation for areal velocity is

• $\frac{\mathrm{dA}}{\mathrm{dt}} \propto \mathrm{\omega r}$

• $\frac{\mathrm{dA}}{\mathrm{dt}} \propto {\mathrm{\omega }}^2\mathrm{r}$

• $\frac{\mathrm{dA}}{\mathrm{dt}} \propto {\mathrm{\omega r}}^2$

• $\frac{\mathrm{dA}}{\mathrm{dt}} \propto \sqrt{\mathrm{\omega r}}$

Two springs of force constant 1000 N/m and 2000 N/m are stretched by same, force. The ratio of their respective potential energies is

• 2 : 1

• 1 : 2

• 4 : 1

• 1 : 4

A particle rests on the top of a hemisphere of radius R. Find the smallest horizontal velocity that must be imparted to the particle, if it is to leave the hemisphere without sliding down it.

• $\sqrt{\mathrm{gR}}$

• $\sqrt{2\mathrm{gR}}$

• $\sqrt{3\mathrm{gR}}$

• $\sqrt{5\mathrm{gR}}$

The moment of inertia of a body about a given axis is 1.2 kg-m². Initially, the body is at rest. In order to produce rotational kinetic energy of 1500 J, angular acceleration of 25 rad/s² must be applied about the axis for a duration of

• 4s

• 2s

• 8s

• 10s