One hollow and one solid cylinder of the same radius roll down on a smooth inclined plane. Then the foot of the inclined plane is reached by 

1.  solid cylinder earlier 

2.  hollow cylinder earlier 

3.  simultaneously both 

4.  the heavier earlier irrespective of being solid or hollow

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. $2(F_1+F_2)$

The motor of an engine is rotating about its axis with an angular velocity of 100 rpm.  It comes to rest is 15 s, after being switched off.  Assuming constant angular deceleration.  What are the numbers of revolutions made by it before coming to rest?

1. 12.5         

2. 40             

3. 32.6           

4. 15.6

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

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

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