Q. No.Three identical spheres, each of mass M, are placed at the corners of a right-angle triangle with mutually perpendicular sides equal to 2m (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.
\({2 \over 3}( \hat{i}+ \hat{j})\)
4.
\({4 \over 3}( \hat{i}+ \hat{j})\)
A meter stick is balanced on a knife edge at its center. When two coins, each of the mass \(5\) gm are put one on top of the other at the \(12.0\) cm mark, the stick is found to be balanced at \(45.0\) cm. What is the mass of the meter stick?
1. \(66\) g
2. \(56\) g
3. \(76\) g
4. \(79\) g
To maintain a rotor at a uniform angular speed of \(200\) rad s-1, an engine needs to transmit a torque of \(180\) N-m. What is the power required by the engine?
1. \(33\) kW
2. \(36\) kW
3. \(28\) kW
4. \(76\) kW
A rope of negligible mass is wound around a hollow cylinder of mass \(3\) kg and radius \(40\) cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of \(30\) N?
(Assume that there is no slipping.)
1. \(21\) rad/s2
2. \(24\) rad/s2
3. \(20\) rad/s2
4. \(25\) rad/s2
The coordinates of the centre of mass of a uniform plate of shape as shown in the figure are:
| 1. | \(\frac{L}{2}, \frac{L}{2} \) | 2. | \(\frac{5 L}{12}, \frac{5 L}{12} \) |
| 3. | \(\frac{5}{3} L, \frac{2}{3} L \) | 4. | \(\frac{3 L}{4}, \frac{L}{2}\) |
A rigid body rotates with an angular momentum of \(L.\) If its kinetic energy is halved, the angular momentum becomes:
1. \(L\)
2. \(L/2\)
3. \(2L\)
4. \(L/\)
A particle is moving with a constant velocity along a line parallel to the positive x-axis. The magnitude of its angular momentum with respect to the origin is:
| 1. | zero |
| 2. | increasing with \(x\) |
| 3. | decreasing with \(x\) |
| 4. | remaining constant |
The center of mass of a system of particles does not depend upon:
| 1. | position of particles |
| 2. | relative distance between particles |
| 3. | masses of particles |
| 4. | force acting on the particle |
The figure below shows two identical particles 1 and 2, each of mass \(m,\) moving in opposite directions with the same speed \(v\) along parallel lines. At a particular instant, \(r_1\) and \(r_2\) are their respective position vectors drawn from point A, which is in the plane of the parallel lines.
Consider the following statements.
| a. | angular momentum \(l_1\) of particle 1 about A is \(l_1=mv(d_1)\) ⊙ |
| b. | angular momentum \(l_1\) of particle 2 about A is \(l_1=mv(r_2)\) ⊙ |
| c. | total angular momentum of the system about A is \(l=mv(r_1+r_2)\) ⊙ |
| d. | total angular momentum of the system about A is \(l=mv(d_2-d_1)\) ⊗ |
Choose the correct option:
| 1. | (a, c) |
| 2. | (a, d) |
| 3. | (b, d) |
| 4. | (b, c) |
The figure shows a lamina in XY-plane. Two axes z and z' pass perpendicular to its plane. A force \(\vec{F}\) acts in the plane of the lamina at point P as shown. (The point P is closer to the z'-axis than the z-axis.)
| a. | torque \(\vec{\tau}\) caused by \(\vec{F}\) about z-axis is along - \(\hat{k}\) |
| b. | torque \(\vec{\tau}'\) caused by \(\vec{F}\) about z'-axis is along - \(\hat{k}\) |
| c. | torque caused by \(\vec{F}\) about the z-axis is greater in magnitude than that about the z'-axis |
| d. | total torque is given by \(\vec{\tau}_{net}=\vec{\tau}+\vec{\tau}'\) |
Choose the correct option:
1. (c, d)
2. (a, c)
3. (b, c)
4. (a, b)
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