A pendulum made of a uniform wire of cross-sectional area \(A\) has time period \(T\). When an additional mass \(M\) is added to its bob, the time period changes to \(T_M\). If the Young’s modulus of the material of the wire is \(Y\) then \(\frac{1}{Y}\) is equal to:
(\(g=\) gravitational acceleration)
1. \( \left[\left(\frac{{T}_{{M}}}{{T}}\right)^2-1\right] \frac{{Mg}}{{A}} \)
2. \(\left[1-\left(\frac{{T}_{{M}}}{{T}}\right)^2\right] \frac{{A}}{{Mg}} \)
3. \(\left[1-\left(\frac{{T}}{{T}_{{M}}}\right)^2\right] \frac{{A}}{{Mg}} \)
4. \(\left[\left(\frac{{T}_{{M}}}{{T}}\right)^2-1\right] \frac{{A}}{{Mg}}\)

Subtopic:  Angular SHM |
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For a simple pendulum, a graph is plotted between its kinetic energy (\(KE\)) and potential energy (\(PE\)) against its displacement \(d\). Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

1. 2.
3.   4.
Subtopic:  Angular SHM |
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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 |
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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 |
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A cylindrical block of wood (density \({=650}~\text{kg m}^{-3}\) ), of base area \(30~\text{cm}^{2}\) and height \({54}~\text{cm},\) floats in a liquid of density \({900}~\text{kg m}^{-3}.\) The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly):
1. \(52~\text{cm}\)
2. \(65~\text{cm}\)
3. \(39~\text{cm}\)
4. \(26~\text{cm}\)
Subtopic:  Simple Harmonic Motion |
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A particle performs simple harmonic motion with amplitude \(A\). Its speed is tripled at the instant that it is at a distance \(\frac{2A}{3}\) from the equilibrium position. The new amplitude of the motion is: 
1. \( \frac{A}{3} \sqrt{41} \)
2. \(3 \mathrm{A} \)
3. \(A \sqrt{3} \)
4. \(\frac{7 A}{3}\)

Subtopic:  Simple Harmonic Motion |
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A particle executes simple harmonic motion with a time period \(T.\) It starts at its equilibrium position at \(t=0.\) How will the graph of its kinetic energy \((KE)\) versus time \((t)\) look like?

1.
2.
3.
4.
Subtopic:  Energy of SHM |
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The ratio of maximum acceleration to maximum velocity in a simple harmonic motion is \(10~\text{s}^{-1}.\) At \({t}=0,\) the displacement is \(5~\text{m}.\) What is the maximum acceleration?
(The initial phase is \(\frac{\pi}{4})\)
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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\(1~\text{kg}\) 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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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. \(1~\text{cm}\)
Subtopic:  Simple Harmonic Motion |
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