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Q. No.The energy of the block is \(E\), and the plane is smooth, the wall at the end \(B\) is smooth. Collisions with walls are elastic. The distance \(AB=l\), the spring is ideal and the spring constant is \(k\). The time period of the motion is:
1.
\(2\pi\sqrt{\dfrac{m}{k}}\)
2.
\(\pi\sqrt{\dfrac{m}{k}}+l\sqrt{\dfrac{2m}{E}}\)
3.
\(2\pi\sqrt{\dfrac{m}{k}}+2l\sqrt{\dfrac{2m}{E}}\)
4.
\(\pi\sqrt{\dfrac{m}{k}}+l\sqrt{\dfrac{m}{2E}}\)
Subtopic: Spring mass system |
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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A particle moves in the x-y plane according to the equation
\(x = A \cos^2 \omega t\) and \(y = A \sin^2 \omega t\)
Then, the particle undergoes:
| 1. | uniform motion along the line \(x + y = A\) |
| 2. | uniform circular motion along \(x^2 + y^2 = A^2\) |
| 3. | SHM along the line \(x + y = A\) |
| 4. | SHM along the circle \(x^2 + y^2 = A^2\) |
Subtopic: Linear SHM |
Level 3: 35%-60%
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A particle of mass \(m\) executes SHM along a straight line with an amplitude \(A\) and frequency \(f.\)
| Assertion (A): | The kinetic energy of the particle undergoes oscillation with a frequency \(2f.\) |
| Reason (R): | Velocity of the particle, \(v = {\dfrac{dx}{dt}}\), its kinetic energy equals \({\dfrac 12}mv^2\) and the particle oscillates sinusoidally with a frequency \(f\). |
| 1. | Both (A) and (R) are True and (R) is the correct explanation of (A). |
| 2. | Both (A) and (R) are True but (R) is not the correct explanation of (A). |
| 3. | (A) is True but (R) is False. |
| 4. | (A) is False but (R) is True. |
Subtopic: Energy of SHM |
52%
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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Two identical blocks are connected by an ideal spring and the system is allowed to oscillate, when undergoing horizontal displacements in opposite directions, with the centre-of-mass at rest. \(O\) is the mid-point of the spring, \(A\) is left end point, \(B\) is the right end-point. The motion of \(A\) is described by: \(x_A = A_0 \sin \omega t\) (displacement is taken to be positive rightward).
Call the mid-point of \(O\) and \(B\) as \(C\) and its \(x\text-\)coordinate as \(x_C.\) Then, the motion of the point \(C\) of the spring is described by:
1. \(x_{C}=A_{0} \sin \left(\omega t+\dfrac{\pi}{2}\right)\)
2. \(x_{C}=\dfrac{A_{0}}{2} \sin \omega t\)
3. \(x_{C}=\dfrac{A_{0}}{2} \sin \left(\omega t+\dfrac{\pi}{2}\right)\)
4. \(x_{C}=\dfrac{A_{0}}{2} \sin (\omega t+\pi)\)
Call the mid-point of \(O\) and \(B\) as \(C\) and its \(x\text-\)coordinate as \(x_C.\) Then, the motion of the point \(C\) of the spring is described by:
1. \(x_{C}=A_{0} \sin \left(\omega t+\dfrac{\pi}{2}\right)\)
2. \(x_{C}=\dfrac{A_{0}}{2} \sin \omega t\)
3. \(x_{C}=\dfrac{A_{0}}{2} \sin \left(\omega t+\dfrac{\pi}{2}\right)\)
4. \(x_{C}=\dfrac{A_{0}}{2} \sin (\omega t+\pi)\)
Subtopic: Spring mass system |
Level 3: 35%-60%
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An elastic ball is projected vertically upward with a speed \(u,\) and it returns to the ground and rebounds, the motion is periodic with a period \(T.\) A simple pendulum, having a length equal to maximum altitude attained by this ball, would have a time period of:
| 1. | \(T\) | 2. | \(\pi T\) |
| 3. | \(\pi\sqrt2T\) | 4. | \(\dfrac{\pi}{\sqrt 2}T\) |
Subtopic: Angular SHM |
53%
Level 3: 35%-60%
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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%
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A particle moves in a plane such that its displacements are the sum of two displacements \(\vec{ r}_1\), and \(\vec{r}_2;\) each of which undergo SHM in opposite phase with respect to the other, but of unequal amplitude. The resultant motion of the particle is:
| 1. | uniform circular motion |
| 2. | elliptical motion |
| 3. | linear SHM |
| 4. | angular SHM along a circle |
Subtopic: Simple Harmonic 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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