A uniform rod of length 1 m and mass 2 kg is suspended by two vertical inextensible strings as shown in following figure. Calculate the tension T (in newtons) in the left string at the instant when the right string snaps (g = 10 m/$s^2$).
       
1. 2.5 N

2. 5 N

3. 7.5 N

4. 10 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?

Let $I_A$ and $I_B$ be moments of inertia of a body about two axes, A and B, respectively. The axis A passes through the centre of mass of the body, but B does not. Which of the following is correct?

1.  $I_A$<$I_B$

2.  If $I_A$<$I_B$, the axes are parallel.

3.  If the axes are parallel, $I_A$<$I_B$

4.  If the axes are not parallel, $I_A$ $\ge$ $I_B$

Three-point masses each of mass 'm', are placed at the vertices of an equilateral triangle of side a. The moment of inertia of the system through a mass m at O and lying in the plane of COA and perpendicular to OA is:

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

A square lamina is as shown in the figure. The moment of inertia of the frame about the three axes is $I_1,$ $I_{\mathrm{2,}}$ $and$ $I_3$ respectively. Select the correct alternative.

     
1. $I_2=I_3>I_1$
2. $I_1>I_2>I_3$
3. $I_2=I_3<I_1$
4. $I_1<I_2<I_3$

Three identical spherical shells, each of mass m and radius r, are placed as shown in the figure. Consider an axis XX’, which is touching two shells and passes through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the XX' axis is:
    
1.$\frac{11}{5}$mr²

2. 3mr²

3. $\frac{16}{5}$mr²

4. 4mr²

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. $\mathrm{\pi } \sec$                 
2. $\frac{3\mathrm{\pi }}{2}\sec$
3. $2\mathrm{\pi } \sec$               
4. $\frac{\mathrm{\pi }}{2}\sec$