A solid cylinder of mass 3 kg is rolling on a horizontal surface with a velocity of 4 ms^-1. It collides with a horizontal spring of force constant 200 Nm^-1. The maximum compression produced in the spring will be:

1. 0.5 m

2. 0.6 m

3. 0.7 m

4. 0.2 m

ABC is an equilateral triangle with O as its centre. F₁, F_2, and F₃ represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero, then the magnitude of F₃ is:
         

1. ${\mathrm{F}}_1+{\mathrm{F}}_2$

2. ${\mathrm{F}}_1-{\mathrm{F}}_2$

3. $\frac{{\mathrm{F}}_1+{\mathrm{F}}_2}{2}$

4. $2\left(\mathrm{F}+{\mathrm{F}}_2\right)$

The moment of inertia of a thin uniform rod of mass M and length L about an axis passing through its mid-point and perpendicular to its length is I₀. Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is:

1.  ${\mathrm{I}}_0+{\mathrm{ML}}^2/4$

2.  ${\mathrm{I}}_0+2{\mathrm{ML}}^2$

3.  ${\mathrm{I}}_0+{\mathrm{ML}}^2$

4.  ${\mathrm{I}}_0+{\mathrm{ML}}^2/2$

A circular disk of a moment of inertia I_t is rotating in a horizontal plane, about its symmetric axis, with a constant angular speed ${\mathrm{\omega }}_{\mathrm{i}}$. Another disk of a moment of inertia I_b is dropped coaxially onto the rotating disk. Initially, the second disk has zero angular speed. Eventually, both the disks rotate with a constant angular speed ${\mathrm{\omega }}_{\mathrm{f}}$. The energy lost by the initially rotating disc due to friction is:

1. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}^2}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^2$

2. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{t}}^2}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^2$

3. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}-{\mathrm{I}}_{\mathrm{t}}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^2$

4. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}{\mathrm{I}}_{\mathrm{t}}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^2$

Two particles that are initially at rest, move towards each other under the action of their mutual attraction. If their speeds are v and 2v at any instant, then the speed of the centre of mass of the system will be:

1. 2v

2. 0

3. 1.5v

4. v

A man of 50 kg mass is standing in a gravity-free space at a height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed of 2 ms^-1. When the stone reaches the floor, the distance of the man above the floor will be:

1. 9.9 m
2. 10.1 m
3. 10 m
4. 20 m

If \(\vec F\) is the force acting on a particle having position vector \(\vec r\) and \(\vec \tau\) be the torque of this force about the origin, then:

A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m are attached gently to the opposite ends of the diameter of the ring, the ring will then rotate with an angular velocity:

1. $\frac{\omega (M-2m)}{M+2m}$

2. $\frac{\omega M}{M+2m}$

3. $\frac{\omega (M+2m)}{M}$

4. $\frac{\omega M}{M+m}$

Two bodies of mass 1 kg and 3 kg have position vectors $\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}} \mathrm{and} -3\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ 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. -$\hat{i}$+$\hat{j}$+$\hat{k}$