Q. No.A smooth inclined plane ends in a vertical circular loop, as shown in figure. A small body is released from height \(h\) as shown. If the body exerts a force of three times its weight on the plane at the highest point of circle then the height \(h =\alpha{R}.\) The value of \(\alpha\) is:
1. \(2\)
2. \(4\)
3. \(3\)
4. \(6\)
1. \(2\)
2. \(4\)
3. \(3\)
4. \(6\)
Subtopic: Conservation of Mechanical Energy |
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A small both \(A\) of mass \(m\) is attached to a massless rigid rod of length \(1~\text{m}\) pivoted at point \(P\) and kept at an angle of \(60^{\circ}\) with vertical as shown in figure. At distance of \(1~\text{m}\) below point \(P\), an identical bob \(B\) is kept at rest on a smooth horizontal surface that extends to a circular track of radius \(R\) as shown in figure. If bob \(B\) just manages to complete the circular path of radius \(R\) upto a point \(Q\) after being hit elastically by bob \(A\), then radius \(R\) is: (in m)
1. \(\dfrac{3}{5}\)
2. \(\dfrac{1}{5}\)
3. \(\dfrac{2+\sqrt{3}}{5}\)
4. \(\dfrac{2-\sqrt{3}}{5}\)
1. \(\dfrac{3}{5}\)
2. \(\dfrac{1}{5}\)
3. \(\dfrac{2+\sqrt{3}}{5}\)
4. \(\dfrac{2-\sqrt{3}}{5}\)
Subtopic: Conservation of Mechanical Energy |
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In case of vertical circular motion of a particle by a thread of length \(r\) if tension in the thread is zero at an angle \(30^\circ\) shown in figure, the velocity at the bottom point \((A)\) of the circular path is: \((g =\) gravitational acceleration\()\)
1. \(\sqrt{5gr}\)
2. \(\sqrt{\dfrac{7}{2} {gr}}\)
3. \(\sqrt{4gr}\)
4.\(\sqrt{\dfrac{5}{2}{gr}}\)
1. \(\sqrt{5gr}\)
2. \(\sqrt{\dfrac{7}{2} {gr}}\)
3. \(\sqrt{4gr}\)
4.\(\sqrt{\dfrac{5}{2}{gr}}\)
Subtopic: Conservation of Mechanical Energy |
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A small mirror of mass \(m\) is suspended by a massless thread of length \(l.\) Then the small angle through which the thread will be deflected when a short pulse of laser of energy \(E\) falls normal on the mirror: (\(c=\)speed of light in vacuum and \(g=\)acceleration due to gravity)
1. \(\theta=\dfrac{E}{m c \sqrt{g l}}\)
2. \(\theta=\dfrac{3E}{4 m c \sqrt{g l}}\)
3. \(\theta=\dfrac{E}{2 m c \sqrt{g l}}\)
4. \(\theta=\dfrac{2E}{mc \sqrt{g l}} \)
1. \(\theta=\dfrac{E}{m c \sqrt{g l}}\)
2. \(\theta=\dfrac{3E}{4 m c \sqrt{g l}}\)
3. \(\theta=\dfrac{E}{2 m c \sqrt{g l}}\)
4. \(\theta=\dfrac{2E}{mc \sqrt{g l}} \)
Subtopic: Conservation of Mechanical Energy |
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A block of mass \(2~\text{kg}\) is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is \(2~\text{m}\) and spring constant is \(200~\text{N/m.} \) The block is pushed such that the length of the spring becomes \(1~\text{m}\) and then released. At distance \({x ~\text m} (x<2) \) from the wall. the speed of the block will be:
1. \(10[1-(2-{x})]^{\frac{3}{2}} ~\text{m/s} \)
2. \(10\left[1-(2-x)^2\right]^{\frac{1}{2}}~\text{m/s}\)
3. \(10\left[1-(2-x)^2\right]~\text{m/s}\)
4. \(10\left[1-(2-x)^2\right]^2 ~\text{m/s} \)
1. \(10[1-(2-{x})]^{\frac{3}{2}} ~\text{m/s} \)
2. \(10\left[1-(2-x)^2\right]^{\frac{1}{2}}~\text{m/s}\)
3. \(10\left[1-(2-x)^2\right]~\text{m/s}\)
4. \(10\left[1-(2-x)^2\right]^2 ~\text{m/s} \)
Subtopic: Conservation of Mechanical Energy |
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A particle is released from height \(S\) above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
1. \(\dfrac{S}{2}, \sqrt{\dfrac{3 g S}{2}} ~\)
2. \(\dfrac{S}{4}, \sqrt{\dfrac{3 g S}{2}} ~\)
3. \(\dfrac{S}{4}, \dfrac{3 g S}{2} ~\)
4. \(\dfrac{S}{2}, \dfrac{3 g S}{2} \)
1. \(\dfrac{S}{2}, \sqrt{\dfrac{3 g S}{2}} ~\)
2. \(\dfrac{S}{4}, \sqrt{\dfrac{3 g S}{2}} ~\)
3. \(\dfrac{S}{4}, \dfrac{3 g S}{2} ~\)
4. \(\dfrac{S}{2}, \dfrac{3 g S}{2} \)
Subtopic: Conservation of Mechanical Energy |
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A body of mass \(100~\text g\) is moving in a circular path of radius \(2~\text m\) on a vertical plane as shown in the figure. The velocity of the body at point \(A\) is \(10~\text{m/s}.\) The ratio of its kinetic energies at point \(B\) and \(C\) is: \((\text{use}~g=9.8~\text{m/s}^{2})\)
| 1. | \(\dfrac{2+\sqrt{2}}{3} \) | 2. | \(\dfrac{3+\sqrt{3}}{2} \) |
| 3. | \(\dfrac{2+\sqrt{3}}{3} \) | 4. | \(\dfrac{3-\sqrt{2}}{2}\) |
Subtopic: Conservation of Mechanical Energy |
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A body of mass \(m\) connected to a massless and unstretchable string goes in vertical circle of radius \(R \) under gravity \(g.\) The other end of the string is fixed at the centre of circle. If velocity at top of circular path is \(n \sqrt{gR} \), where, \(n>1 \), then ratio of kinetic energy of the body at bottom to that at top of the circle is:
1. \(\dfrac{n+4}{n}\)
2. \(\dfrac{n^2}{n^2+4}\)
3. \(\dfrac{n}{n+4}\)
4. \(\dfrac{n^2+4}{n^2}\)
1. \(\dfrac{n+4}{n}\)
2. \(\dfrac{n^2}{n^2+4}\)
3. \(\dfrac{n}{n+4}\)
4. \(\dfrac{n^2+4}{n^2}\)
Subtopic: Conservation of Mechanical Energy |
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A bob of mass m is suspended at a point \(O\) by a light string of length l and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity \(v_0\) at the point A the string becomes slack when, the bob reaches at the point \(D\). The ratio of the kinetic energy of the bob at the points \(B\) and \(C\) is _____.
1. \(3\)
2. \(1\)
3. \(4\)
4. \(2\)
1. \(3\)
2. \(1\)
3. \(4\)
4. \(2\)
Subtopic: Conservation of Mechanical Energy |
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A body of
(given, \(R=14\) m, \(g=10\) m/s2 and \(\sqrt{2}=1.4\))
1. \(21.9\) m/s
2. \(10.6\) m/s
3. \(19.8\) m/s
4. \(16.7\) m/s
(given, \(R=14\) m, \(g=10\) m/s2 and \(\sqrt{2}=1.4\))
1. \(21.9\) m/s
2. \(10.6\) m/s
3. \(19.8\) m/s
4. \(16.7\) m/s
Subtopic: Conservation of Mechanical Energy |
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