A thin wire of length L and uniform linear mass density $\rho$ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX' is

(1) $\frac{\rho L^3}{8{\pi }^2}$

(2) $\frac{\rho L^3}{16{\pi }^2}$

(3) $\frac{5 \rho L^3}{16{\pi }^2}$

(4) $\frac{3\rho L^3}{8{\pi }^2}$

The one-quarter sector is cut from a uniform circular disc of radius R. This sector has mass M. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

(1) $\frac{1}{2}M{R}^2$

(2) $\frac{1}{4}M{R}^2$

(3) $\frac{1}{8}M{R}^2$

(4) $\sqrt{2}M{R}^2$

A uniform rod of length 2L is placed with one end in contact with the horizontal and is then inclined at an angle $\alpha$ to the horizontal and allowed to fall without slipping at contact point. When it becomes horizontal, its angular velocity will be

(1) $\omega =\sqrt{\frac{3g \sin \alpha }{2L}}$

(2) $\omega =\sqrt{\frac{2L}{3g \sin \alpha }}$

(3) $\omega =\sqrt{\frac{6g \sin \alpha }{L}}$

(4) $\omega =\sqrt{\frac{L}{g \sin \alpha }}$

A billiard ball of mass m and radius r, when hit in a horizontal direction by a cue at a height h above its centre, acquired a linear velocity $v_0$ . The angular velocity ${\omega }_0$ acquired by the ball is :

(1) $\frac{5v_0{r}^2}{2h}$

(2) $\frac{2v_0{r}^2}{5h}$

(3) $\frac{2v_{0}h}{5r^2}$

(4) $\frac{5v_{0}h}{2r^2}$

A mass \(m\) hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass \(m\) and radius \(R\). Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass \(m\), if the string does not slip on the pulley, is:
1. \(\dfrac{3}{2}g\)
2. \(g\)
3. \(\dfrac{2}{3}g\)
4. \(\dfrac{g}{3}\)

A ladder is leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the trace of its center of mass?

(1) 

(2)

(3)

(4)

A carpet of mass m made of inextensible material is rolled along its length in the form of a cylinder of radius r and kept on a rough floor. The decrease in the potential energy of the system, when the carpet is unrolled to a radius $\frac{r}{2}$ without sliding is (g = acceleration due to gravity)

(1) $\frac{3}{4}$mgr

(2) $\frac{5}{8}mgr$

(3) $\frac{7}{8}mgr$

(4) $\frac{1}{2}mgr$

The moment of inertia of a uniform circular disc of radius 'R' and mass 'M' about an axis touching the disc at its edge and normal to the disc is :

(1) $\frac{3}{2}M{R}^2$

(2) $\frac{1}{2}M{R}^2$

(3) $M{R}^2$

(4) $\frac{2}{5}M{R}^2$

A solid disc rolls clockwise without slipping over a horizontal path with constant speed v. Then the magnitude of the velocities of points A, B and C with respect to the standing observer are respectively:

(1) v,v and v

(2) 2v, $\sqrt{2}$v and zero

(3) 2v, 2v and zero

(4) 2v, $\sqrt{2}$v and $\sqrt{2}$v

The total torque about pivot A provided by the forces shown in the figure, for L = 3.0 m, is

(1) 210 Nm

(2) 140 Nm

(3) 95 Nm

(4) 75 Nm