A body of mass M is moving on a circular track of radius r in such a way that its kinetic energy K depends on the distance travelled by the body s according to relation K = $\beta$s, where $\beta$ is a constant. The angular acceleration of the body is:

1.  $\frac{\beta r}{M^2}$

2.  $\sqrt{\frac{\beta r}{M}}$

3.  $\frac{M{r}^2}{\beta }$

4.  $\frac{\beta }{Mr}$

If a particle moves in a circle with a constant angular speed ($\mathrm{\omega }$) about point O, then its angular speed about point A will be:
           
1.  2$\mathrm{\omega }$

2.  $\frac{\mathrm{\omega }}{2}$

3.  $\mathrm{\omega }$

4.  $\frac{\mathrm{\omega }}{4}$

Which of the following is the value of the torque of force \(F\) about origin \(O:\)

1. \(\vec{\tau}=5(1-\sqrt{3}) \hat{k}\) N-m
2. \(\vec{\tau}=5(1-\sqrt{3}) \hat{j}\) N-m
3. \(\vec{\tau}=5(\sqrt{3}-1) \hat{i}\) N-m
4. \(\vec{\tau}=\sqrt{3} \hat{j}\) N-m

Four thin rods, each of mass m and length L, form a square. The moment of inertia on any side of the square is:

             
1.  $\frac{5}{3}m{L}^2$

2.  $4$ $m{l}^2$

3.  $\frac{m{L}^2}{4}$

4.   $\frac{2}{3}m{L}^2$

In the three figures, each wire has a mass M, radius R and a uniform mass distribution. If they form part of a circle of radius R, then about an axis perpendicular to the plane and passing through the centre (shown by crosses), their moment of inertia is in the order:

1.  $I_A$ $>$ $I_B$ $>$ $$ $I_C$

2.  $I_A$ $=$ $I_B$ $=$ $I_C$

3.  $I_A$ $<$ $I_B$ $<$ $I_C$

4.  $I_A$ $<$ $I_C$ $<$ $I_B$

A solid body rotates about a stationary axis according to the equation $\theta = 6t - 2t^3$. What is the average angular velocity over the time interval between t = 0 and the time when the body comes to rest? ($\theta$ : angular displacements, t : time)

The value of M, as shown, for which the rod will be in equilibrium is:
      

Particles A and B are separated by 10 m, as shown in the figure. If A is at rest and B started moving with a speed of 20 m/s then the angular velocity of B with respect to A at that instant is:

A uniform cubical block of side L rests on a rough horizontal surface with coefficient of friction $\mu$. A horizontal force F is applied on the block as shown. If there is sufficient friction between the block and the ground, then the torque due to normal reaction about its centre of mass is:

1.  $Zero$

2.  $FL$

3.  $\frac{FL}{2}$

4.  $\frac{3FL}{2}$