Q. No.The mass per unit length of a non-uniform rod of length \(L\) is given by \(\mu =λx^{2}\) where \(\lambda\) is a constant and \(x\) is the distance from one end of the rod.
The distance between the centre of mass of the rod and this end is:
1. \(\frac{L}{2}\)
2. \(\frac{L}{4}\)
3. \(\frac{3L}{4}\)
4. \(\frac{L}{3}\)
1. \(\frac{L}{2}\)
2. \(\frac{L}{4}\)
3. \(\frac{3L}{4}\)
4. \(\frac{L}{3}\)
At \(t=0\), the positions of the two blocks are shown. There is no external force acting on the system. Find the coordinates of the center of mass of the system at \(t=3\) seconds:
| 1. | \((1,0)\) | 2. | \((3,0)\) |
| 3. | \((4.5,0)\) | 4. | \((2.25,0)\) |
A ring and a disc have the same mass and roll without slipping at the same linear velocity v. If the total kinetic energy of the ring is 8 J, then the total kinetic energy of the disc will be:
1. 8 J
2. 6 J
3. 16 J
4. 4 J
A solid homogenous sphere is moving on a rough horizontal surface, partially rolling and partially sliding. During this motion of this sphere:
1. The total kinetic energy is conserved.
2. The angular momentum of the sphere about the point of contact is conserved
3. Only the rotational kinetic energy about the centre of mass is conserved
4. Angular momentum about the centre of mass is conserved
A uniform square plate \(ABCD\) has a mass of \(10\) kg.
If two point masses of \(5\) kg each are placed at the corners \(C\) and \(D\) as shown in the adjoining figure, then the centre of mass shifts to the mid-point of:
1. \(OH\)
2. \(DH\)
3. \(OG\)
4. \(OF\)
A string of negligible thickness is wrapped several times around a cylinder kept on a rough horizontal surface. A man standing at a distance l from the cylinder holds one end of the string and pulls the cylinder towards him. There is no slipping anywhere. The length of the string that passed through the hand of the man while the cylinder reaches his hands is-[Assume radius of the cylinder is negligible compared to length 'l' of string]
| 1. | l | 2. | 2l |
| 3. | 3l | 4. | 4l |
Four-point masses each of value \(m\) are placed at the corners of a square ABCD of side \(l\). The moment of inertia of this system about an axis passing through A and parallel to BD will be:
| 1. | \(2ml^2\) | 2. | \(4ml^2\) |
| 3. | \(3ml^2\) | 4. | \(ml^2\) |
A particle rotating on a circular path of the radius \(\frac{4}{\pi}~\text{m}\) at \(300\) rpm reaches \(600\) rpm in \(6\) revolutions. If the angular velocity increases at a constant rate, find the tangential acceleration of the particle:
1. \(10\) m/s2
2. \(12.5\) m/s2
3. \(25\) m/s2
4. \(50\) m/s2
A rod is falling down with constant velocity \(V_0\) as shown. It makes contact with hinge A and rotates around it. The angular velocity of the rod just after the moment when it comes in contact with hinge A is:
| 1. | \(2 \mathrm{V}_0 / 3 \mathrm{L} \) | 2. | \(3 \mathrm{V}_0 / 2 \mathrm{L} \) |
| 3. | \(\mathrm{V}_0 / \mathrm{L} \) | 4. | \(2 \mathrm{V}_0 / 5 \mathrm{L}\) |
Two loads \(P_1\) \(P_2\)\((P_1>P_2)\) are connected by a string passing over a fixed pulley. The center of gravity of loads are initially at the same height. Find the acceleration of the center of gravity of the system:
1. \(\left(\frac{(P_1-P_2)^{\frac{1}{2}}}{P_1+P_2}\right)g\)
2. \(\left(\frac{P_1-P_2}{P_1+P_2}\right)g\)
3. \(\left( \frac{P_1-P_2}{P_1+P_2}\right)^2g\)
4. \(\left( \frac{P_1+P_2}{P_1-P_2}\right)g\)
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