Shown in the figure is a rigid and uniform one-metre long rod, AB, held in the horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass 'm' and has another weight of mass '2 m' hung at a distance of 75 cm from A. The tension in the string at A is:
   
1.  2 mg
2.  0.5 mg
3.  0.75 mg
4. 1 mg

Consider two uniform discs of the same thickness and different radii \(R_1=R\) and \(R_2=\alpha R\)  made of the same material. If the ratio of their moments of inertia, \(I_1\) and\(I_2,\) respectively, about their axes is \(I_1:I_2=1:16,\)then the value of \(\alpha\) is:
1. \(\sqrt{2}\)
2. \(4\)
3. \(2\)
4. \(2\sqrt{2}\)

Four identical solid spheres each of mass 'm' and radius 'a' are placed with their centres on the four corners of a square of side 'b'. The moment of inertia of the system about one side of the square where the axis of rotation is parallel to the plane of the square is :

1. $\frac{4}{5}{\mathrm{ma}}^2+2{\mathrm{mb}}^2$

2. $\frac{8}{5}m{a}^2+m{b}^2$

3. $\frac{8}{5}{\mathrm{ma}}^2+2{\mathrm{mb}}^2$

4. $\frac{4}{5}m{a}^2$

A force \(\vec{F}=\hat{i}+2\hat{j}+3\hat{k}~\text{N}\) acts at a point \(\hat{4i}+3\hat{j}-\hat{k}~\text{m}\). Let the magnitude of the torque about the point \(\hat{i}+2\hat{j}+\hat{k}~\text{m}\) be \(\sqrt{x}~\text{N-m}\). The value of \(x\) is:

From a circular ring of mass \({M}\) and radius \(R\), an arc corresponding to a \(90^\circ\)$$sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is \(K\) times \(MR^2\). The value of \(K\) will be:
1. \(\frac{1}{4}\)
2. \(\frac{1}{8}\)
3. \(\frac{3}{4}\)
4. \(\frac{7}{8}\)

Two gear wheels that are meshed together have radii of 0.50 cm and 0.15 cm. The number of revolutions made by the smaller one when the larger one goes through 3 revolutions is:

1. 5 revolutions 

2. 20 revolutions 

3. 1 revolution

4. 10 revolutions

A wheel has an angular acceleration of 3.0 rad/s² and an initial angular speed of 2.00 rad/s. In a time of 2 s, it has rotated through an angle (in radian) of:
1. 6
2. 10
3. 12
4. 4

If a rod of length 3 m with its mass acting per unit length, is directly proportional to distance x from one of its ends, then its centre of gravity from that end will be at: 

For a body, with angular velocity \( \vec{\omega }=\hat{i}-2\hat{j}+3\hat{k}\)  and radius vector \( \vec{r }=\hat{i}+\hat{j}++\hat{k},\)  its velocity will be:
1. \(-5\hat{i}+2\hat{j}+3\hat{k}\)
2. \(-5\hat{i}+2\hat{j}-3\hat{k}\)
3. \(-5\hat{i}-2\hat{j}+3\hat{k}\)
4. \(-5\hat{i}-2\hat{j}-3\hat{k}\)

A circular disc is to be made by using iron and aluminium so that it acquires a maximum moment of inertia about its geometrical axis. It is possible with: