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Q. No.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. \(4kA^2\)
1. \({\Large\frac{1}{2}}kA^2\)
2. \(kA^2\)
3. \(2kA^2\)
4. \(4kA^2\)
Subtopic: Energy of SHM |
Level 4: Below 35%
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A block of mass \(m\) is attached to two fixed light springs of identical stiffness \(k,\) and the block rests on the \(x\text-y\) plane. The springs are along the \(x\) & \(y\) axes. The system is viewed from above. The block undergoes small oscillations along a line which makes \(45^\circ\) with the \(x\text-\)axis. The angular frequency of these oscillations is:
1. \(\sqrt{\Large\frac{2k}{m}}\)
2. \(\sqrt{\Large\frac{\sqrt2k}{m}}\)
3. \(\sqrt{\Large\frac{2\sqrt2k}{m}}\)
4. \(\sqrt{\Large\frac{k}{m}}\)
1. \(\sqrt{\Large\frac{2k}{m}}\)
2. \(\sqrt{\Large\frac{\sqrt2k}{m}}\)
3. \(\sqrt{\Large\frac{2\sqrt2k}{m}}\)
4. \(\sqrt{\Large\frac{k}{m}}\)
Subtopic: Combination of Springs |
Level 4: Below 35%
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Which, of the following, represents the displacement in simple harmonic motion?
| (A) | \(x=A\sin^2\omega t\) |
| (B) | \(x=A\sin\omega t+B\cos2\omega t\) |
| (C) | \(x=A\sin^2\omega t+B\cos2\omega t\) |
| 1. | A only | 2. | A and B |
| 3. | A and C | 4. | A, B and C |
Subtopic: Types of Motion |
Level 4: Below 35%
Hints
A spring 'lengthens' by \(l\) when a block of mass \(m\) is suspended from it. This block is suspended from the same spring and the system is allowed to oscillate vertically, in space where the gravity is \(\bigg({\large\frac19}\bigg)^{\text{th}}\) of its original value. The time period of small oscillations is:
(where \(g\) is the acceleration due to gravity on the surface of the Earth)
| 1. | \(2\pi{\sqrt{\large\frac{l}{g}}}\) | 2. | \(6\pi{\sqrt{\large\frac{l}{g}}}\) |
| 3. | \(2\pi{\sqrt{\large\frac{9l}{8g}}}\) | 4. | \(2\pi{\sqrt{\large\frac{3l}{g}}}\) |
Subtopic: Spring mass system |
Level 4: Below 35%
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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 |
Level 4: Below 35%
Hints
Two SHMs given by their displacements along the respective directions are superposed: \(x=A\sin\omega t~~~~(1^{\text{st}}~\text{SHM along }x\text{-axis})\\ y=A\sin\bigg(\omega t+{\large\frac{\pi}{2}}\bigg)~~~~(2^{\text{nd}}~\text{SHM along }y\text{-axis}).\)
The resultant motion is:
The resultant motion is:
| 1. | SHM along a straight line |
| 2. | SHM along a circular arc |
| 3. | uniform circular motion |
| 4. | motion along an elliptic path |
Subtopic: Types of Motion |
Level 3: 35%-60%
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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 |
Level 3: 35%-60%
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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 \(80\text{ km/h},\) while on the return trip the trains travel at twice that speed. The services are maintained round the clock. Trains leave station \(A\) every \(30\text{ min}\) for station \(B\) and reach \(B\) in \(2\text{ hrs.}\) 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 |
Level 3: 35%-60%
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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 |
Level 3: 35%-60%
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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 |
Level 3: 35%-60%
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