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Q. No.Two particles undergo SHM along the same straight line, moving with the same frequency and amplitude but with a phase difference of \(60^\circ\) between each other. If they have a maximum speed of \(v_0,\) the maximum relative velocity between them will be:
| 1. | \(v_0\) | 2. | \(2v_0\) |
| 3. | \({\dfrac{\sqrt3}{2}}v_0\) | 4. | \(\sqrt3v_0\) |
Subtopic: Phasor Diagram |
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Two identical masses are connected by a spring of spring constant \(k,\) and the individual masses are observed to undergo SHM with their centre of mass remaining at rest. The amplitude of oscillation of one of the masses is \(A.\) The total energy of oscillation is:
| 1. | \({\Large\frac{1}{2}}kA^2\) | 2. | \(kA^2\) |
| 3. | \(2kA^2\) | 4. | \(4kA^2\) |
Subtopic: Energy of SHM |
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Which, of the following, represents the displacement in simple harmonic motion?
1. A only
2. A and B
3. A and C
4. A, B and C
| (A) | \(x=A\sin^2\omega t\) |
| (B) | \(x=A\sin\omega t+B\cos2\omega t\) |
| (C) | \(x=A\sin^2\omega t+B\cos2\omega t\) |
2. A and B
3. A and C
4. A, B and C
Subtopic: Types of Motion |
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A uniform rod of length \(l\) is suspended by an end and is made to undergo small oscillations. The time period of small oscillation is \(T\). Then, the acceleration due to gravity at this place is:
| 1. | \(4\pi^2\dfrac{l}{T^2}\) | 2. | \(\dfrac{4\pi^2}{3}\dfrac{l}{T^2}\) |
| 3. | \(\dfrac{8\pi^2}{3}\dfrac{l}{T^2}\) | 4. | \(\dfrac{12\pi^2l}{T^2}\) |
Subtopic: Angular SHM |
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A light rod \(AB\) is hinged at \(A\) so that it is free to rotate about \(A.\) It is initially horizontal with a small block of mass \(m\) attached at \(B,\) and a spring (constant - \(k\)) holding it vertically up at its mid-point. The time period of vertical oscillations of the system is:
| 1. | \(2 \pi \sqrt{\dfrac{m}{k}} \) | 2. | \(\pi \sqrt{\dfrac{m}{k}} \) |
| 3. | \(4\pi \sqrt{\dfrac{m}{k}}\) | 4. | \(\dfrac{\pi}{2} \sqrt{\dfrac{m}{k}}\) |
Subtopic: Spring mass system |
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Given below are two statements:
| Statement I: | If the acceleration of a particle is directed towards a fixed point, and proportional to the distance from that point – the motion is SHM. |
| Statement II: | During SHM, the kinetic energy of the particle oscillates at twice the frequency of the SHM. |
| 1. | Statement I is incorrect and Statement II is correct. |
| 2. | Both Statement I and Statement II are correct. |
| 3. | Both Statement I and Statement II are incorrect. |
| 4. | Statement I is correct and Statement II is incorrect. |
Subtopic: Energy of SHM |
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A particle undergoes SHM with an amplitude of \(10\) cm and a time period of \(4\) s. The average velocity of the particle during the course of its motion from its mean position to its extreme position is:
1. \(5\) cm/s
2. \(10\) cm/s
3. at least \(10\) cm/s
4. at most \(10\) cm/s
1. \(5\) cm/s
2. \(10\) cm/s
3. at least \(10\) cm/s
4. at most \(10\) cm/s
Subtopic: Simple Harmonic Motion |
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An elastic ball rebounds vertically to a height \(h\) above the ground, the period of its motion will be:
| 1. | \(\begin{aligned} \large\sqrt\frac{2h}{g} & \\ \end{aligned}\) | 2. | \(\begin{aligned} \large\sqrt\frac{8h}{g} & \\ \end{aligned}\) |
| 3. | \(\begin{aligned} \large\sqrt\frac{h}{2g} & \\ \end{aligned}\) | 4. | \(\begin{aligned} 2\large{\sqrt\frac{h}{g}} & \\ \end{aligned}\) |
Subtopic: Types of Motion |
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Trains travel between station \(A\) and station \(B\): on the way up (from \(A~\text{to}~B\)) - they travel at a speed of \(80\) km/h, while on the return trip the trains travel at twice that speed. The services are maintained round the clock. Trains leave station \(A\) every \(30\) min for station \(B\) and reach \(B\) in \(2\) hrs. All trains operate continuously, without any rest at \(A\) or \(B\).
| 1. | the frequency of trains leaving \(B\) must be twice as much as \(A\). |
| 2. | the frequency of trains leaving \(B\) must be half as much as \(A\). |
| 3. | the frequency of trains leaving \(B\) is equal to that at \(A\). |
| 4. | the situation is impossible to maintain unless larger number of trains are provided at \(A\). |
Subtopic: Types of Motion |
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A small block of mass \(m\) slides a distance \(L\) down a smooth incline and rebounds elastically back up. The period of the motion is: (using standard symbols where necessary)
1. \(\Large\sqrt{\frac{4L\sin\theta}{g}}\)
2. \(\Large\sqrt{\frac{8L\sin\theta}{g}}\)
3. \(\Large\sqrt{\frac{8L}{g\sin\theta}}\)
4. \(\Large\sqrt{\frac{4L}{g\sin\theta}}\)
1. \(\Large\sqrt{\frac{4L\sin\theta}{g}}\)
2. \(\Large\sqrt{\frac{8L\sin\theta}{g}}\)
3. \(\Large\sqrt{\frac{8L}{g\sin\theta}}\)
4. \(\Large\sqrt{\frac{4L}{g\sin\theta}}\)
Subtopic: Linear SHM |
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