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Q. No.A \(1~\text{kg}\) block attached to a spring vibrates with a frequency of \(1~\text{Hz}\) on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an \(8~\text{kg}\) block placed on the same table. So, the frequency of vibration of the \(8~\text{kg}\) block is:
1. \(\frac{1}{4}~\text{Hz}\)
2. \(\dfrac{1}{2\sqrt2}~\text{Hz}\)
3. \(2~\text{Hz}\)
4. \(\dfrac{1}{2}~\text{Hz}\)
Subtopic: Spring mass system |
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Level 2: 60%+
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In an experiment to determine the period of a simple pendulum of length \(1~\text{m},\) it is attached to different spherical bobs of radii \({r}_1\) and \({r}_2.\) The two spherical bobs have uniform mass distribution. If the relative difference in the periods, is found to be \(5\times10^{-4}~\text{s},\) the difference in radii, \(|{r}_1-{r}_2|\) is best given by:
1. \(0.1~\text{cm}\)
2. \(0.01~\text{cm}\)
3. \(0.5~\text{cm}\)
4. \(1~\text{cm}\)
1. \(0.1~\text{cm}\)
2. \(0.01~\text{cm}\)
3. \(0.5~\text{cm}\)
4. \(1~\text{cm}\)
Subtopic: Simple Harmonic Motion |
Level 3: 35%-60%
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A body of mass \(M\) and a charge \(q\) is connected to a spring of spring constant \(k.\) It oscillates along the \(x\text{-direction}\) about its equilibrium position, taken to be at \(x = 0,\) with an amplitude \(A.\) An electric field \(E\) is applied along the \(x\text{-direction}. \)
Which of the following statements is correct?
Which of the following statements is correct?
| 1. | The total energy of the system is \(\dfrac{1}{2}m\omega^2A^2+\dfrac{1}{2}\dfrac{q^2E^2}{k}.\) |
| 2. | The new equilibrium position is at a distance \(\dfrac{2qE} {k}\) from \(x = 0.\) |
| 3. | The new equilibrium position is at a distance \(\dfrac{qE} {2k}\) from \(x = 0.\) |
| 4. | The total energy of the system is \(\dfrac{1}{2}m\omega^2A^2-\dfrac{1}{2}\dfrac{q^2E^2}{k}.\) |
Subtopic: Energy of SHM |
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Two masses \({m}~\text{and}~ \frac {{m}}{2}\) are connected at the two ends of a massless rigid rod of length \(l.\) The rod is suspended by a thin wire of torsional constant \(k\) at the center of mass of the rod-mass system (see figure). Because of the torsional constant \(k,\) the restoring torque is \(\tau=k \theta\) for angular displacement \(\theta.\) If the rod is rotated by \(\theta_0\) and released, the tension in it when it passes through its mean position will be:
1. \(\dfrac{3k\theta_0^2}{l}\)
2. \(\dfrac{2k\theta_0^2}{l}\)
3. \(\dfrac{k\theta_0^2}{l}\)
4. \(\dfrac{k\theta_0^2}{2l}\)
1. \(\dfrac{3k\theta_0^2}{l}\)
2. \(\dfrac{2k\theta_0^2}{l}\)
3. \(\dfrac{k\theta_0^2}{l}\)
4. \(\dfrac{k\theta_0^2}{2l}\)
Subtopic: Angular SHM |
Level 3: 35%-60%
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A block of mass \(0.1~\text{kg}\) is connected to an elastic spring constant \(640~\text{Nm}^{-1}\) and oscillates in a damping medium of damping constant \(10^{-2}~\text{kg s}^{-1}.\) The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value is closest to:
1. \(2~\text{s}\)
2. \(3.5~\text{s}\)
3. \(7~\text{s}\)
4. \(5~\text{s}\)
1. \(2~\text{s}\)
2. \(3.5~\text{s}\)
3. \(7~\text{s}\)
4. \(5~\text{s}\)
Level 4: Below 35%
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In an engine the piston undergoes vertical simple harmonic motion with amplitude \(7~\text{cm}.\) A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer longer stays in contact with the piston is:
1. \(0.7~\text{Hz}\)
2. \(1.2~\text{Hz}\)
3. \(1.9~\text{Hz}\)
4. \(0.1~\text{Hz}\)
1. \(0.7~\text{Hz}\)
2. \(1.2~\text{Hz}\)
3. \(1.9~\text{Hz}\)
4. \(0.1~\text{Hz}\)
Subtopic: Simple Harmonic Motion |
Level 4: Below 35%
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An oscillator of mass \(M\) is at rest in the equilibrium position in a potential \(V = \frac {1} {2} k(x – X)^2 .\) A particle of mass \(m\) comes from the right with speed \(u\) and collides completely inelastically with \(M\) and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after \(13\) collisions is:
(Take \(M = 10\), \(m = 5,\) \(u = 1, \) \(k = 1\))
1. \(\dfrac{2}{3}\)
2. \(\dfrac{1}{\sqrt3}\)
3. \(\sqrt {\dfrac{3}{5}} \)
4. \(\dfrac{1}{2}\)
(Take \(M = 10\), \(m = 5,\) \(u = 1, \) \(k = 1\))
1. \(\dfrac{2}{3}\)
2. \(\dfrac{1}{\sqrt3}\)
3. \(\sqrt {\dfrac{3}{5}} \)
4. \(\dfrac{1}{2}\)
Subtopic: Energy of SHM |
Level 4: Below 35%
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A particle executes simple harmonic motion and is located at \(x = a, b\) and \(c\) at times \(t_0, 2t_0\) and \(3t_0\) respectively. The frequency of the oscillation is:
1. \(\frac{1}{2\pi t_0}\cos^{-1}\bigg(\frac{a+c}{2b}\bigg) \)
2. \(\frac{1}{2\pi t_0}\cos^{-1}\bigg(\frac{a+2b}{3c}\bigg)\)
3. \(\frac{1}{2\pi t_0}\cos^{-1}\bigg(\frac{a+b}{2c}\bigg)\)
4. \(\frac{1}{2\pi t_0}\cos^{-1}\bigg(\frac{2a+3c}{b}\bigg)\)
1. \(\frac{1}{2\pi t_0}\cos^{-1}\bigg(\frac{a+c}{2b}\bigg) \)
2. \(\frac{1}{2\pi t_0}\cos^{-1}\bigg(\frac{a+2b}{3c}\bigg)\)
3. \(\frac{1}{2\pi t_0}\cos^{-1}\bigg(\frac{a+b}{2c}\bigg)\)
4. \(\frac{1}{2\pi t_0}\cos^{-1}\bigg(\frac{2a+3c}{b}\bigg)\)
Subtopic: Simple Harmonic Motion |
Level 3: 35%-60%
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The time period of a simple pendulum is \({T}\) inside a lift when the lift is stationary. If the lift moves upwards with an acceleration \(\frac{g}{2},\) the time period of the pendulum will be:
1. \(\sqrt{\dfrac{2}{3}}~ {T}\)
2. \(\sqrt3~{T}\)
3. \(\dfrac{{T}}{{\sqrt3}}\)
4. \(\sqrt{\dfrac{3}{2}} ~{T}\)
1. \(\sqrt{\dfrac{2}{3}}~ {T}\)
2. \(\sqrt3~{T}\)
3. \(\dfrac{{T}}{{\sqrt3}}\)
4. \(\sqrt{\dfrac{3}{2}} ~{T}\)
Subtopic: Simple Harmonic Motion |
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Level 1: 80%+
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A particle is executing simple harmonic motion \(\text{(SHM)}\) of amplitude \({(A)},\) along the \(x\text-\)axis about, \({x=0.}\) When its potential energy \({(PE)}\) equals kinetic energy \({(KE)},\) the position of the particle will be:
1. \(\dfrac{A}{2}\)
2. \(\dfrac{A}{2\sqrt{2}}\)
3. \(\dfrac{A}{\sqrt{2}}\)
4. \({A}\)
1. \(\dfrac{A}{2}\)
2. \(\dfrac{A}{2\sqrt{2}}\)
3. \(\dfrac{A}{\sqrt{2}}\)
4. \({A}\)
Subtopic: Energy of SHM |
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Level 1: 80%+
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