Q. No.A mass \(m\) moves in a circle on a smooth horizontal plane with velocity \(v_0\) at a radius \(R_0.\) The mass is attached to a string that passes through a smooth hole in the plane, as shown in the figure.
The tension in the string is increased gradually and finally, \(m\) moves in a circle of radius \(\frac{R_0}{2}.\) The final value of the kinetic energy is:
1.
\( m v_0^2 \)
2.
\( \dfrac{1}{4} m v_0^2 \)
3.
\( 2 m v_0^2 \)
4.
\( \dfrac{1}{2} m v_0^2\)
The tension in the string is increased gradually and finally, \(m\) moves in a circle of radius \(\frac{R_0}{2}.\) The final value of the kinetic energy is:
Three identical spherical shells, each of mass \(m\) and radius \(r\) are placed as shown in the figure. Consider an axis \(XX',\) which is touching two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the \(XX'\) axis is:
| 1. | \(\dfrac{11}{5}mr^2\) | 2. | \(3mr^2\) |
| 3. | \(\dfrac{16}{5}mr^2\) | 4. | \(4mr^2\) |
The ratio of the acceleration for a solid sphere (mass \(m\) and radius \(R\)) rolling down an incline of angle \(\theta\) without slipping and slipping down the incline without rolling is:
1. \(5:7\)
2. \(2:3\)
3. \(2:5\)
4. \(7:5\)
A rod \(PQ\) of mass \(M\) and length \(L\) is hinged at end \(P\). The rod is kept horizontal by a massless string tied to point \(Q\) as shown in the figure. When the string is cut, the initial angular acceleration of the rod is:
| 1. | \(\dfrac{g}{L}\) | 2. | \(\dfrac{2g}{L}\) |
| 3. | \(\dfrac{2g}{3L}\) | 4. | \(\dfrac{3g}{2L}\) |
Two persons of masses \(55~\text{kg}\) and \(65~\text{kg}\) respectively, are at the opposite ends of a boat. The length of the boat is \(3.0~\text{m}\) and weighs \(100~\text{kg}.\) The \(55~\text{kg}\) man walks up to the \(65~\text{kg}\) man and sits with him. If the boat is in still water, the centre of mass of the system shifts by:
1. \(3.0~\text{m}\)
2. \(2.3~\text{m}\)
3. zero
4. \(0.75~\text{m}\)
A solid cylinder of mass \(3\) kg is rolling on a horizontal surface with a velocity of \(4\) ms-1. It collides with a horizontal spring of force constant \(200\) Nm-1. The maximum compression produced in the spring will be:
1. \(0.5\) m
2. \(0.6\) m
3. \(0.7\) m
4. \(0.2\) m
\(\mathrm{ABC}\) is an equilateral triangle with \(O\) as its centre. \(F_1,\) \(F_2,\) and \(F_3\) represent three forces acting along the sides \({AB},\) \({BC}\) and \({AC}\) respectively. If the total torque about \(O\) is zero, then the magnitude of \(F_3\) is:
1. \(F_1+F_2\)
2. \(F_1-F_2\)
3. \(\frac{F_1+F_2}{2}\)
4. \(2F_1+F_2\)
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. \(v\)
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. | \(\vec r\cdot\vec \tau=0\text{ and }\vec F\cdot\vec \tau\neq0\) |
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. | \(\sqrt{2}:\sqrt{3}\) |
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