Q. No.A uniform disc of radius \(R\) and mass \(M\) is free to oscillate about the axis \(A\) as shown in the figure. For small oscillations the time period is:
(\(g\) is acceleration due to gravity)
1. \(2 \pi \sqrt{\dfrac{5 R}{4 g}}~\)
2. \(2 \pi \sqrt{\dfrac{2 R}{3 g}}~\)
3. \(2 \pi \sqrt{\dfrac{3 R}{2 g}}~\)
4. \(2 \pi \sqrt{\dfrac{3 R}{g}}~\)
(\(g\) is acceleration due to gravity)
1. \(2 \pi \sqrt{\dfrac{5 R}{4 g}}~\)
2. \(2 \pi \sqrt{\dfrac{2 R}{3 g}}~\)
3. \(2 \pi \sqrt{\dfrac{3 R}{2 g}}~\)
4. \(2 \pi \sqrt{\dfrac{3 R}{g}}~\)
Subtopic: Simple Harmonic Motion |
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Level 3: 35%-60%
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Match List-I with List-II.
Choose the correct answer from the options given below:
1. \(\mathrm{A\text-III, B\text-I, C\text-IV, D\text-II }\)
2. \(\mathrm{A\text-II, B\text-I, C\text-III, D\text-IV }\)
3. \(\mathrm{A\text-III, B\text-II, C\text-IV, D\text-I}\)
4. \(\mathrm{A\text-II, B\text-I, C\text-IV, D\text-III }\)
| List-I | List-II | ||
| \(\mathrm{(A)}\) | \(\sin ^2 \omega t\) | \(\mathrm{(I)}\) | Periodic with time period \(T=\dfrac{\pi}{\omega}\) but not simple harmonic motion (\(\mathrm{SHM}\)) |
| \(\mathrm{(B)}\) | \(\sin ^3(2 \omega t)\) | \(\mathrm{(II)}\) | Periodic with time period \(T=\dfrac{2\pi}{\omega}\) but Not \(\mathrm{SHM}\) |
| \(\mathrm{(C)}\) | \(\sin (\omega t)+\cos (\pi \omega t)\) | \(\mathrm{(III)}\) | Periodic with time period \(T=\dfrac{\pi}{\omega}\) and \(\mathrm{SHM}\) |
| \(\mathrm{(D)}\) | \(\cos \omega t+\cos 2 \omega t\) | \(\mathrm{(IV)}\) | Non-periodic |
1. \(\mathrm{A\text-III, B\text-I, C\text-IV, D\text-II }\)
2. \(\mathrm{A\text-II, B\text-I, C\text-III, D\text-IV }\)
3. \(\mathrm{A\text-III, B\text-II, C\text-IV, D\text-I}\)
4. \(\mathrm{A\text-II, B\text-I, C\text-IV, D\text-III }\)
Subtopic: Simple Harmonic Motion |
Level 3: 35%-60%
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The velocity of a particle executing simple harmonic motion along \(x\text{-axis}\) is described as \(v^2=50-x^2\), where \(x\) represents displacement. If the time period of motion is \(\dfrac{x}{7} ~\text{s}\), the value of \(x\) is:
1. \(44\)
2. \(50\)
3. \(60\)
4. \(80\)
1. \(44\)
2. \(50\)
3. \(60\)
4. \(80\)
Subtopic: Simple Harmonic Motion |
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Level 2: 60%+
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A cylindrical block of mass \(M\) and area of cross section \(A\) is floating in a liquid of density \(\rho\) and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is:
1. \(2 \pi \sqrt{\dfrac{M}{\rho A g}}\)
2. \(\pi \sqrt{\dfrac{2{M}}{\rho {Ag}}}\)
3. \(\pi \sqrt{\dfrac{\rho {A}}{{Mg}}}\)
4. \(2\pi \sqrt{\dfrac{\rho {A}}{{Mg}}}\)
1. \(2 \pi \sqrt{\dfrac{M}{\rho A g}}\)
2. \(\pi \sqrt{\dfrac{2{M}}{\rho {Ag}}}\)
3. \(\pi \sqrt{\dfrac{\rho {A}}{{Mg}}}\)
4. \(2\pi \sqrt{\dfrac{\rho {A}}{{Mg}}}\)
Subtopic: Simple Harmonic Motion |
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Level 1: 80%+
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Given below are two statements:
Choose the correct answer from the options given below:
| Assertion (A): | Knowing initial position \(x_0\) and initial momentum \(p_0\) is enough to determine the position and momentum at any time \(t\) for a simple harmonic motion with a given angular frequency \(\omega.\) |
| Reason (R): | The amplitude and phase can be expressed in terms of \(x_0\) and \(p_0\). |
Choose the correct answer from the options given below:
| 1. | (A) is false, but (R) is true |
| 2. | (A) is true, but (R) is false |
| 3. | Both (A) and (R) are true, but (R) is NOT the correct explanation of (A) |
| 4. | Both (A) and (R) are true, and (R) is the correct explanation of (A) |
Subtopic: Simple Harmonic Motion |
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Level 2: 60%+
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A light hollow cube of side length \(10~\text{cm}\) and mass \(10~\text g,\) is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is \(y\pi \times10^{-2}~\text s,\) where the value of \(y\) is (Acceleration due to gravity, \(g=10~\text{m/s}^2,\) density of water =\(10^3~\text{kg/m}^3)\)
1. \(2\)
2. \(4\)
3. \(6\)
4. \(1\)
1. \(2\)
2. \(4\)
3. \(6\)
4. \(1\)
Subtopic: Simple Harmonic Motion |
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A particle is executing simple harmonic motion with time period \(2~ \text s\) and amplitude \(1~\text{cm}.\) If \(D\) and \(d\) are the total distance and displacement covered by the particle in \(12.5~\text s,\) then \(\dfrac{D}{d}\) is:
1. \(\dfrac{15}{4}\)
2. \(\dfrac{16}{5}\)
3. \(10\)
4. \(25\)
1. \(\dfrac{15}{4}\)
2. \(\dfrac{16}{5}\)
3. \(10\)
4. \(25\)
Subtopic: Simple Harmonic Motion |
80%
Level 1: 80%+
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Given below are two statements:
In the light of the above statements, choose the most appropriate answer from the options given below:
| Assertion (A): | Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain. |
| Reason (R): | Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa. |
| 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: Simple Harmonic Motion |
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Level 2: 60%+
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Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
In the light of the above statements, choose the correct answer from the options given below:
| Assertion (A): | A simple pendulum is taken to a planet of mass and radius, \(4\) times and \(2\) times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet. |
| Reason (R): | The mass of the pendulum remains unchanged at Earth and the other planet. |
| 1. | Both (A) and (R) are true but (R) is NOT the correct explanation of (A) |
| 2. | Both (A) and (R) are true and (R) is the correct explanation of (A) |
| 3. | (A) is false but (R) is true |
| 4. | (A) is true but (R) is false |
Subtopic: Simple Harmonic Motion |
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Level 2: 60%+
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The displacement of a particle executing Simple Harmonic Motion is given by \(x=10 \sin \left(\omega t+\dfrac{\pi}{3}\right)~\text m \). The time period of motion is \(3.14~\text s.\) The velocity of the particle at \(t=0 \) is:
1. \(10~\text{m/s}\)
2. \(17.3~\text{m/s}\)
3. \(5~\text{m/s}\)
4. \(20~\text{m/s}\)
1. \(10~\text{m/s}\)
2. \(17.3~\text{m/s}\)
3. \(5~\text{m/s}\)
4. \(20~\text{m/s}\)
Subtopic: Simple Harmonic Motion |
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Level 2: 60%+
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