A uniform beam \(3.0~\text m\) long, of weight \(100~\text N\) has a \(300~\text N\) weight placed \(0.5~\text m\) from one end. The beam is suspended by a string \(1.0~\text m\) from the same end. A diagram of the weights placed on the beam is given below:
            
How far from the other end must a weight of \(80~\text N\) be placed for the beam to be balanced?

1.	\(0.75~\text m\)	2.	\(2.25~\text m\)
3.	\(1.25~\text m\)	4.	\(1.875~\text m\)

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D${}_1$ has a 2 kg mass, 0.2 m radius, and an initial angular velocity of 50 rad s${}^{-1}$. Disc D${}_2$ has 4 kg mass, 0.1 m radius, and initial angular velocity of 200 rad s${}^{-1}$. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad.s${}^{-1}$) of the system will be:

1. 60

2. 100

3. 120

4. 40

When a mass is rotating in a plane about a fixed point, its angular momentum is directed along:

A circular platform is mounted on a frictionless vertical axle. Its radius is R = 2m and its moment of inertia about the axle is 200 kg m². Initially, it is at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at a speed of 1 m s^–1 relative to the ground. The time taken by the man to complete one revolution is:
1. $\mathrm{\pi }<mspace\; width="1em"></mspace>\sec$                 
2. $\frac{3\mathrm{\pi }}{2}\sec$
3. $2\mathrm{\pi }<mspace\; width="1em"></mspace>\sec$               
4. $\frac{\mathrm{\pi }}{2}\sec$

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 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 ladder is leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the path followed by its center of mass?

1.		2.
3.		4.

A uniform rod of length \(1~\text m\) and mass \(2~\text {kg}\) is suspended by two vertical inextensible strings as shown in the following figure. The tension \(T\) (in newtons) in the left string at the instant when the right string snaps is:
\((g = 10~\text{m/s}^ 2 ).\)
             
1. \(2.5~\text N\)
2. \(5~\text N\)
3. \(7.5~\text N\)
4. \(10~\text N\)

A man '\(A\)', mass \(60\) kg, and another man '\(B\)', mass \(70\) kg, are sitting at the two extremes of a \(2\) m long boat, of mass \(70\) kg, standing still in the water as shown. They come to the middle of the boat. (Neglect friction). How far does the boat move on the water during the process?