Q. No.Two particles that are initially at rest, move towards each other under the action of their mutual attraction. If their speeds are \(v\) and \(2v\) at any instant, then the speed of the centre of mass of the system will be:
1. \(2v\)
2. \(0\)
3. \(1.5v\)
4. \(v\)
1. \(2v\)
2. \(0\)
3. \(1.5v\)
4. \(v\)
A man of \(50\) kg mass is standing in a gravity-free space at a height of \(10\) m above the floor. He throws a stone of \(0.5\) kg mass downwards with a speed of \(2\) ms-1. When the stone reaches the floor, the distance of the man above the floor will be:
1. \(9.9\) m
2. \(10.1\) m
3. \(10\) m
4. \(20\) m
If \(\vec F\) is the force acting on a particle having position vector \(\vec r\) and \(\vec \tau\) be the torque of this force about the origin, then:
| 1. | \(\vec r\cdot\vec \tau\neq0\text{ and }\vec F\cdot\vec \tau=0\) |
| 2. | \(\vec r\cdot\vec \tau>0\text{ and }\vec F\cdot\vec \tau<0\) |
| 3. | \(\vec r\cdot\vec \tau=0\text{ and }\vec F\cdot\vec \tau=0\) |
| 4. | \(\vec r\cdot\vec \tau=0\text{ and }\vec F\cdot\vec \tau\neq0\) |
A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m are attached gently to the opposite ends of the diameter of the ring, the ring will then rotate with an angular velocity:
| 1. | \(\frac{\omega(M-2 m)}{M+2 m} \) | 2. | \(\frac{\omega M}{M+2 m} \) |
| 3. | \(\frac{\omega(M+2 m)}{M} \) | 4. | \(\frac{\omega M}{M+m}\) |
Two bodies of mass \(1\) kg and \(3\) kg have position vectors \(\hat{i}+2\hat{j}+\hat{k}\) and \(-3\hat{i}-2\hat{j}+\hat{k}\) respectively. The centre of mass of this system has a position vector:
1. \(-2\hat{i}+2\hat{k}\)
2. \(-2\hat{i}-\hat{j}+\hat{k}\)
3. \(2\hat{i}-\hat{j}-2\hat{k}\)
4. \(-\hat{i}+\hat{j}+\hat{k}\)
2. \(\frac{2}{3}Ml^2\)
3. \(\frac{13}{3}Ml^2\)
4. \(\frac{1}{3}Ml^2\)
The ratio of the radii of gyration of a circular disc to that of a circular ring, each of the same mass and radius, around their respective axes is:
| 1. | \(\sqrt{3}:\sqrt{2}\) | 2. | \(1:\sqrt{2}\) |
| 3. | \(\sqrt{2}:1\) | 4. | \(\sqrt{2}:\sqrt{3}\) |
A thin rod of length \(L\) and mass \(M\) is bent at its midpoint into two halves so that the angle between them is \(90^{\circ}\). The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is:
1. \(\frac{ML^2}{24}\)
2. \(\frac{ML^2}{12}\)
3. \(\frac{ML^2}{6}\)
4. \(\frac{\sqrt{2}ML^2}{24}\)
A wheel has an angular acceleration of \(3.0\) rad/s2 and an initial angular speed of \(2.0\) rad/s. In a time of \(2\) s, it has rotated through an angle (in radians) of:
| 1. | \(6\) | 2. | \(10\) |
| 3. | \(12\) | 4. | \(4\) |
A uniform rod \(AB\) of length \(l\) and mass \(m\) is free to rotate about point \(A\). The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about \(A\) is \(\dfrac{ml^2}{3}\) the initial angular acceleration of the rod will be:
1. \(\dfrac{2g}{3l}\)
2. \(\dfrac{mgl}{2}\)
3. \(\dfrac{3}{2}gl\)
4. \(\dfrac{3g}{2l}\)
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