The bricks, each of length L and mass M, are arranged as shown from the wall. The distance of the centre of mass of the system from the wall is :

(1) L/4

(2) L/2

(3) (3/2) L

(4) (11/12) L

A uniform disk of mass M and radius R is mounted on a fixed horizontal axis. A block of mass m hangs from a massless string that is wrapped around the rim of the disk. The magnitude of the acceleration of the falling block (m) is :

(1) $\frac{2M}{M+2m}g$

(2) $\frac{2m}{M+2m}g$

(3) $\frac{M+2m}{2M}g$

(4) $\frac{2M+m}{2M}g$

A horizontal force F is applied such that the block remains stationary. Then which of the following statement is false?

(1) f = mg [where f is the friction force]

(2) F = N [where N is the normal force]

(3) F will not produce torque

(4) N will not produce torque

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

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

                   
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. \(\dfrac{3\rho L^3}{8\pi^2}\)

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: 
                  

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, 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 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) $\frac{g}{3}$