Q. No.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)
Consider the following statements.
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)
A uniform cube of mass \(m\) and side \(a\) is placed on a frictionless horizontal surface. A vertical force \(F\) is applied to the edge as shown in the figure. Match the following (most appropriate choice).
| List- I | List- II | ||
| (a) | \(mg/4<F<mg/2\) | (i) | cube will move up. |
| (b) | \(F>mg/2\) | (ii) | cube will not exhibit motion. |
| (c) | \(F>mg\) | (iii) | cube will begin to rotate and slip at A. |
| (d) | \(F=mg/4\) | (iv) | normal reaction effectively at \(a/3\) from A, no motion. |
| 1. | a - (i), b - (iv), c - (ii), d - (iii) |
| 2. | a - (ii), b - (iii), c - (i), d - (iv) |
| 3. | a - (iii), b - (i), c - (ii), d - (iv) |
| 4. | a - (i), b - (ii), c - (iv), d - (iii) |
The angular speed of the wheel of a vehicle is increased from \(360~\text{rpm}\) to \(1200~\text{rpm}\) in \(14\) seconds. Its angular acceleration will be:
| 1. | \(2\pi ~\text{rad/s}^2\) | 2. | \(28\pi ~\text{rad/s}^2\) |
| 3. | \(120\pi ~\text{rad/s}^2\) | 4. | \(1 ~\text{rad/s}^2\) |
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)
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
In the figure given below, O is the centre of an equilateral triangle ABC and are three forces acting along the sides AB, BC and AC. What should be the magnitude of so that total torque about O is zero?
1.
2.
3.
4. Not possible
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