Q. No.| 1. | \(2v\) | 2. | zero |
| 3. | \(v\) | 4. | \(4v\) |
| 1. | \(5~ \text m\) | 2. | \(\frac{10}{3} \mathrm{~m}\) |
| 3. | \(\frac{20}{3} \mathrm{~m}\) | 4. | \(10~ \text m\) |
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. \(\frac{4}{3}(\hat{i}+\hat{j})\)
Which of the following statements are correct?
| a. | centre of mass of a body always coincides with the centre of gravity of the body . |
| b. | centre of gravity of a body is the point about which the total gravitational torque on the body is zero. |
| c. | a couple on a body produce both translational and rotation motion in a body. |
| d. | mechanical advantage greater than one means that small effort can be used to lift a large load. |
| 1. | (a) and (b) | 2. | (b) and (c) |
| 3. | (c) and (d) | 4. | (b) and (d) |
Two persons of masses \(55\) kg and \(65\) kg respectively, are at the opposite ends of a boat. The length of the boat is \(3.0\) m and weighs \(100\) kg. The \(55\) kg man walks up to the \(65\) kg man and sits with him. If the boat is in still water, the centre of mass of the system shifts by:
1. \(3.0\) m
2. \(2.3\) m
3. zero
4. \(0.75\) m
Three masses are placed on the \(x\)-axis: \(300~\text{g}\) at origin, \(500~\text{g}\) at \(x= 40~\text{cm}\) and \(400~\text{g}\) at \(x= 70~\text{cm}\). The distance of the centre of mass from the origin is:
1. \(45~\text{cm}\)
2. \(50~\text{cm}\)
3. \(30~\text{cm}\)
4. \(40~\text{cm}\)
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. \(v\)
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
Given the following statements:
| 1. | The centre of gravity (C.G.) of a body is the point at which the weight of the body acts. |
| 2. | If the earth is assumed to have an infinitely large radius, the centre of mass coincides with the centre of gravity. |
| 3. | To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G. |
| 4. | The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis. |
Which one of the following pairs of statements is correct?
1. (a) and (b)
2. (b) and (c)
3. (c) and (d)
4. (d) and (a)
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