A ladder is leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the path followed by 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 diameter and normal to the disc will be:
1. \(\frac{3}{2} M R^{2}\)
2. \(\frac{1}{2} M R^{2}\)
3. \(M R^{2}\)
4. \(\frac{2}{5} M R^{2}\)${\mathrm{}}^{}$

An automobile engine develops 100 kW when rotating at a speed of 1800 rev/min. What torque does it deliver ?

1. 350 N-m

2. 440 N-m

3. 531 N-m

4. 628 N-m

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) $\frac{3}{2}g$

(2) $g$

(3) $\frac{2}{3}g$

(4) $\frac{g}{3}$

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, acquires a linear velocity $v_0$ . The angular velocity ${\omega }_0$ acquired by the ball will be:

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

The one-quarter sector is cut from a uniform circular disc of radius \(R\). This sector has a 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 will be: 
                  

                   
1. \(\dfrac{\rho L^3}{8\pi^2}\)
2. \(\dfrac{\rho L^3}{16\pi^2}\)
3. \(\dfrac{5\rho L^3}{16\pi^2}\)
4. \(\dfrac{3\rho L^3}{8\pi^2}\)