The moment of inertia of a thin uniform circular disc about one of its diameter is I. Its moment of inertia about an axis perpendicular to the circular surface and passing through its center will be:

1. $\sqrt{2}I$

2. 2 l

3. $\frac{I}{2}$

4. $\frac{I}{\sqrt{2}}$

A child is standing on the edge of a merry-go-round that has the shape of a disk, as shown in the figure. The mass of the child is 40 kilograms. The merry-go-round has a mass of 200 kilograms and a radius of 2.5 meters, and it is rotating with an angular velocity of $\omega =2.0$ radians per second. The child then walks slowly towards the center of the merry-go-round. When the child reaches the center, what is the angular velocity of the disc? (The size of the child can be neglected.)
     
1. 2.0 rad/s

2. 2.2 rad/s

3. 2.4 rad/s

4. 2.8 rad/s

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?

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

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, $I_A$ $\ge$ $I_B$

A horizontal heavy uniform bar of weight \(W\) is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to:

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. $I_1<I_2<I_3$