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.  $\frac{\beta }{Mr}$

For \(L = 3.0~\text{m,}\) the total torque about pivot \(A\) provided by the forces as shown in the figure is:

A force \(- F \hat k\) acts on point \(O\) the origin of the coordinate system. What is the torque about the point \((1,-1)\)?
1. \(-F(\hat i + \hat j)\)
2. \(F(\hat i + \hat j)\)
3. \(-F(\hat i -\hat j)\)
4. \(F(\hat i - \hat j)\)

A wheel with a radius of \(20\) cm has forces applied to it as shown in the figure. The torque produced by the forces of \(4\) N at \(A\), \(8~\)N at \(B\), \(6\) N at \(C\), and \(9~\)N at \(D\), at the angles indicated, is:

1. \(5.4\) N-m anticlockwise

2. \(1.80\) N-m clockwise

3. \(2.0\) N-m clockwise

4. \(3.6\) N-m clockwise

On a rough horizontal surface (coefficient of friction $\mu$), a cubical block of side 'a' and mass m is projected horizontally. The net torque on the block about its centre of mass till the block stops is equal to:

1.  zero

2.  $\frac{1}{2}\mu mga$

3.  $\mu$mga

4.  mga

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

In the figure given below, \(O\) is the centre of an equilateral triangle \(ABC\) and \(\vec{F_{1}} ,\vec F_{2}, \vec F_{3}\) are three forces acting along the sides \(AB\), \(BC\) and \(AC\). What should be the magnitude of \(\vec{F_{3}}\) so that total torque about \(O\) is zero?

1. \(\left|\vec{F_{3}}\right|= \left|\vec{F_{1}}\right|+\left|\vec{F_{2}}\right|\)
2. \(\left|\vec{F_{3}}\right|= \left|\vec{F_{1}}\right|-\left|\vec{F_{2}}\right|\)
3. \(\left|\vec{F_{3}}\right|= \vec{F_{1}}+2\vec{F_{2}}\)
4. Not possible

A force \(\vec{F}=\hat{i}+2\hat{j}+3\hat{k}~\text{N}\) acts at a point \(\hat{4i}+3\hat{j}-\hat{k}~\text{m}\). Let the magnitude of the torque about the point \(\hat{i}+2\hat{j}+\hat{k}~\text{m}\) be \(\sqrt{x}~\text{N-m}\). The value of \(x\) is:
1. \(145\)
2. \(195\)
3. \(245\)
4. \(295\)

A horizontal heavy uniform bar of weight \(W\) is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to: