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Q. No.A simple harmonic oscillator of angular frequency \(2~\text{rad s}^{-1}\) is acted upon by an external force \({F}=\sin t~\text{ N}. \) If the oscillator is at rest in its equilibrium at \({t}=0,\) its position at later times is proportional to:
1. \(\cos{t}-\frac{1}{2}\sin 2t \)
2. \(\sin{t}+\frac{1}{2}\cos2{t} \)
3. \(\sin{t}+\frac{1}{2}\sin2{t} \)
4. \(\sin{t}-\frac{1}{2}\sin2{t} \)
Subtopic: Simple Harmonic Motion |
Level 3: 35%-60%
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\(x\) and \(y\) displacements of a particle are given as \({x(t)} =\text{a}\sin\omega {t}\) and \({y(t)}={a}\sin 2\omega {t}.\) Its trajectory will look like:
| 1. |  |
3. |  |
| 2. |  |
4. |  |
Subtopic: Types of Motion |
Level 3: 35%-60%
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Two particles are performing simple harmonic motion in a straight line about the same equilibrium point. The amplitude and time period for both particles are the same and equal to \({A}\) and \({T},\) respectively. At time \({t = 0}\) one particle has displacement \(A\) while the other one has displacement \(\frac{-A}{2}\) and they are moving towards each other. If they cross each other at a time \({t},\) then the value of \({t}\) is:
1. \({T\over 4}\)
2. \({5T\over 6}\)
3. \({T\over 3}\)
4. \({T\over 6}\)
1. \({T\over 4}\)
2. \({5T\over 6}\)
3. \({T\over 3}\)
4. \({T\over 6}\)
Subtopic: Simple Harmonic Motion |
Level 4: Below 35%
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A particle performs simple harmonic motion (SHM). The ratio of its maximum acceleration to its maximum velocity is \(10~\text{s}^{-1}.\) At \({t}=0,\) the displacement of the particle is \(5~\text{m},\) and the initial phase is \(\dfrac{\pi}{4}.\) What is the maximum acceleration of the particle?
1. \(500\sqrt2~\text{m/s}^2\)
2. \(500~\text{m/s}^2\)
3. \(750~\text{m/s}^2\)
4. \(750\sqrt2~\text{m/s}^2\)
1. \(500\sqrt2~\text{m/s}^2\)
2. \(500~\text{m/s}^2\)
3. \(750~\text{m/s}^2\)
4. \(750\sqrt2~\text{m/s}^2\)
Subtopic: Simple Harmonic Motion |
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Level 2: 60%+
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A simple pendulum made of a bob of mass \(m\) and a metallic wire of negligible mass has time period \(2~\text{s}\) at \(T = 0^\circ\text{C.}\) If the temperature of the wire is increased and the corresponding charge in its time period is plotted against its temperature, the resulting graph is a line of slope \({S}.\) If the coefficient of linear expansion of metal is \(\alpha\) then the value of \({S}\) is:
1. \(\mathrm{1\over \alpha}\)
2. \(2\alpha\)
3. \(\alpha\over 2\)
4. \(\alpha\)
1. \(\mathrm{1\over \alpha}\)
2. \(2\alpha\)
3. \(\alpha\over 2\)
4. \(\alpha\)
Subtopic: Simple Harmonic Motion |
Level 4: Below 35%
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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. \(\dfrac{1}{2}~\text{Hz}\)
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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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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Level 2: 60%+
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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. \(5~\text{s}\)
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