ABC is an equilateral triangle with O as its cente. ${\overrightarrow{F}}_1, \overrightarrow{F_2} and \overrightarrow{F_3}$ represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then the magnitude of $\overrightarrow{F_3}$ is

1. $F_1+F_2$

2. $F_1-F_2$

3. $\frac{F_1+F_2}{2}$

4. $2(F_1+F_2)$

$If F be a force acting on a particle having the position vector r and \tau be the torque of the force about the origin, then;$
$\left(a\right) r.\tau =0 and F.\tau =0 \left(b\right) r.\tau =0 and F.\tau \ne 0$
$\left(c\right) r.\tau \ne 0 and F.\tau \ne 0 \left(d\right)r.\tau \ne 0 and F.\tau =0$ 

A T joint is formed by two identical rods A and B each of mass m and length L in the XY plane as shown. Its moment of inertia about axis passing through A and perpendicular to the plane of the joint,

1.  $\frac{2{\mathrm{mL}}^2}{3}$

2.  $\frac{{\mathrm{mL}}^2}{12}$

3.  $\frac{{\mathrm{mL}}^2}{6}$

4.  None of these

The moment of inertia of a loop of radius R and mass M, about any tangent line in its plane will be

1. $\frac{3M{R}^2}{2}$

2. $\frac{M{R}^2}{2}$

3. $M{R}^2$

4. $\frac{M{R}^2}{4}$

A hollow sphere of diameter 0.2 m and mass 2 kg is rolling on an inclined plane with velocity v = 0.5 m/ s. The kinetic energy of the sphere is

1.  0.1 J

2.  0.3 J

3.  0.5 J

4.  0.42 J

A man of the mass M stands at one end of a plank of length L which lies at rest on a frictionless surface. The man walks to the other end of the plank. If the mass of the plank is $\frac{\mathrm{M}}{3}$, the distance that the man moves relative to the ground is

1. $\frac{3\mathrm{L}}{4}$

2. $\frac{\mathrm{L}}{4}$

3. $\frac{4\mathrm{L}}{5}$

4. $\frac{\mathrm{L}}{3}$

Ratio of total kinetic energy and rotational kinetic energy in the motion of a disc is

1. 1:1

2. 2:7

3. 1:2

4. 3:1

A cylinder is rolling over a surface. Which points on it move rectilinearly?

1. All points on the curved surface of the cylinder.

2. All points on the flat surfaces of the cylinder.

3. All points on the axis of the cylinder.

4. None of the above

A uniform rod AB of length l and mass m is free to rotate about point A. The rod is released from rest in horizontal position. Given that the moment of inerita of the rod about A is $\frac{{\mathrm{ml}}^2}{3}$ the initial angular acceleration of the rod will be
                         

$1. \frac{2\mathrm{g}}{3\mathrm{l}}$
$2. \mathrm{mg}\frac{\mathrm{l}}{2}$
$3. \frac{3}{2}\mathrm{gl}$
$4. \frac{3\mathrm{g}}{2\mathrm{l}}$