If $\overrightarrow{\mathrm{F}}$ is the force acting on a particle having position vector $\overrightarrow{\mathrm{r}}$ and $\overrightarrow{\mathrm{\tau }}$ be the torque of this force about the origin, then 

(a) $\overrightarrow{\mathrm{r}}.\overrightarrow{\mathrm{\tau }}\ne 0 \mathrm{and} \overrightarrow{\mathrm{F}}.\overrightarrow{\mathrm{\tau }}=0$                  

(b) $\overrightarrow{\mathrm{r}}.\overrightarrow{\mathrm{\tau }}>0 \mathrm{and} \overrightarrow{\mathrm{F}}.\overrightarrow{\mathrm{\tau }}<0$

(c) $\overrightarrow{\mathrm{r}}.\overrightarrow{\mathrm{\tau }}=0 \mathrm{and} \overrightarrow{\mathrm{F}}.\overrightarrow{\mathrm{\tau }}=0$

(d) $\overrightarrow{\mathrm{r}}.\overrightarrow{\mathrm{\tau }}=0 \mathrm{and} \overrightarrow{\mathrm{F}}.\overrightarrow{\mathrm{\tau }}\ne 0$

Two bodies of mass 1kg and 3kg have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$, respectively. The centre of mass of this system has a position vector -

(1) $-2\hat{i}+2\hat{k}$

(2) $-2\hat{i}-\hat{j}+\hat{k}$

(3) $2\hat{i}-\hat{j}-2\hat{k}$

(4) $-\hat{i}+\hat{j}+\hat{k}$

Four identical thin rods each of mass M and length t, form a square frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is 

(1) $\frac{4}{3}M{t}^2$                                   

(2) $\frac{2}{3}M{t}^2$

(3) $\frac{13}{3}M{t}^2$                                 

(4) $\frac{1}{3}M{t}^2$

A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere?

1. Angular velocity 

2. Moment of inertia

3. Angular momentum 

4. Rotational kinetic energy

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. \(\dfrac{3\rho L^3}{8\pi^2}\)