Q. No.A child sits stationary at one end of a long trolley moving uniformly with a speed \(v\) on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, then the speed of the centre of mass of the (trolley + child) system:
1. decreases
2. increases
3. remains unchanged
4. none of these
| Assertion (A): | The center of mass of an isolated system of particles remains at rest if it is initially at rest. |
| Reason (R): | Internal forces acting within a system cannot change the velocity of the center of mass which is proportional to the total momentum of the system. |
| 1. | (A) is True but (R) is False. |
| 2. | (A) is False but (R) is True. |
| 3. | Both (A) and (R) are True and (R) is the correct explanation of (A). |
| 4. | Both (A) and (R) are True but (R) is not the correct explanation of (A). |
1. \(0.4\) m from \(m_{A}\)
2. \(0.6\) m from \(m_{A}\)
3. \(0.5\) m from \(m_{A}\)
4. \(0.7\) m from \(m_{A}\)
| 1. | \((0,0)\) | 2. | \((0,3)\) |
| 3. | \((3,0)\) | 4. | \((2,3)\) |
| 1. | \(\dfrac{m_1}{m_2}\) | 2. | \(\sqrt{\dfrac{m_1}{m_2}}\) |
| 3. | \(\dfrac{m_2}{m_1}\) | 4. | \(\sqrt{\dfrac{m_2}{m_1}}\) |
| 1. | 0.25 m below the table | 2. | 0.5 m below the table |
| 3. | 0.33 m below the table | 4. | 0.4 m below the table |
1. \(5~ \text m\)
2. \(\dfrac{10}{3} \mathrm{~m}\)
3. \(\dfrac{20}{3} \mathrm{~m}\)
4. \(10~ \text m\)
1. \((a / 2, a / 2)\)
2. \(\text { (a/ } \sqrt{2}, a / \sqrt{2})\)
3. \(\left(\dfrac{\sqrt{2} a}{3}, \dfrac{\sqrt{2} a}{3}\right)\)
4. \(\left(\dfrac{a}{3}, \dfrac{a}{3}\right)\)
Two particles of mass \(5~\text{kg}\) and \(10~\text{kg}\) respectively are attached to the two ends of a rigid rod of length \(1~\text{m}\) with negligible mass. The centre of mass of the system from the \(5~\text{kg}\) particle is nearly at a distance of:
1. \(50~\text{cm}\)
2. \(67~\text{cm}\)
3. \(80~\text{cm}\)
4. \(33~\text{cm}\)
Three identical spheres, each of mass \(M\), are placed at the corners of a right-angle triangle with mutually perpendicular sides equal to \(2~\text{m}\) (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin, find the position vector of the centre of mass.
1. \(2(\hat{i}+\hat{j})\)
2. \(\hat{i}+\hat{j}\)
3. \(\frac{2}{3}(\hat{i}+\hat{j})\)
4. \(\frac{4}{3}(\hat{i}+\hat{j})\)
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