Assertion (A): For large angle in simple pendulum \(T > 2\pi \sqrt{\frac{l}{g}}\)
Reason (R): \(sin \theta < \theta\), if the restoring force. \(mg sin \theta\) is replaced by \(mg \theta\), this amounts to effective reduction in g for large angle, hence an increase in T.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true: The period of a simple pendulum increases for large amplitudes compared to the small angle approximation. Reason (R) is true: For large angles, \(sin \theta < \theta\). This makes the actual restoring force smaller than the linear approximation, effectively reducing the 'g' and thereby increasing the period \(T\). (R) correctly explains (A).
Assertion (A): We can assume damped oscillation to be approximately periodic motion for small damping
Reason (R): Small damping means \( \frac{b}{\sqrt{km}} \ll 1 \)
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true: For small damping, the amplitude decays slowly, and the frequency is nearly constant, making the motion approximately periodic. Reason (R) is true: Small damping is characterized by a small damping factor \(b\) relative to \(\sqrt{km}\), specifically \( \frac{b}{\sqrt{km}} \ll 1\) (or \( \zeta ll 1\)). This condition directly ensures the motion is approximately periodic.
Assertion (A): When a simple pendulum is made to oscillate on the surface of moon, its time period increases.
Reason (R): Gravity at moon is less than gravity at earth.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true: The period of a simple pendulum is inversely proportional to the square root of 'g' (\(T = 2pi sqrt{l/g}\)). Reason (R) is true: Gravity on the moon is significantly less than on Earth. Since 'g' decreases, the period 'T' increases, making (R) the correct explanation for (A).
Assertion (A): \(x = \sin^2(\omega t)\) represents a SHM about mean position \(x = \frac{1}{2}\).
Reason (R): \(a \propto -x\) is the necessary condition for SHM.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A): \(x = \sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2}\). Let \(y = x - \frac{1}{2} = -\frac{1}{2}\cos(2\omega t)\). This is SHM about \(x = \frac{1}{2}\). So (A) is true. Reason (R): For SHM, acceleration is proportional to negative displacement \(a = -\omega^2 x\). So (R) is true. However, (R) does not explain (A).
Assertion (A): If PE of a particle executing SHM is given by \(U = x^2 – 10x + 27\), then it is executing SHM about \(x = 5\).
Reason (R): At mean position, restoring force is zero.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A): Given \(U = x^2 - 10x + 27 = (x-5)^2 + 2\). For SHM, \(U = \frac{1}{2}k(x-x_0)^2 + U_0\). Comparing, mean position \(x_0 = 5\). So (A) is true. Reason (R): The restoring force \(F = -\frac{dU}{dx}\). At equilibrium (mean) position, \(F=0\). So (R) is true. (R) does not explain (A).
Assertion (A): In resonance amplitude is infinity, in presence of dissipative forces.
Reason (R): At resonance driving frequency is equal to natural frequency of the system.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is false. In the presence of dissipative forces, the amplitude at resonance is finite, not infinite. Reason (R) is true. At resonance, the driving frequency matches the natural frequency of the system. Thus, (A) is false and (R) is true.
Assertion (A): In damped oscillation, the motion is periodic.
Reason (R): In damped oscillation, the amplitude decreases due to dissipative forces.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is false. Damped oscillation is not strictly periodic because its amplitude continuously decreases with time. Reason (R) is true. The amplitude in damped oscillations decreases due to the energy loss caused by dissipative forces. Thus, (A) is false and (R) is true.
Assertion (A): The amplitude of damped oscillation depends on damping constants.
Reason (R): The angular frequency for a damped oscillation depends on damping constant only.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true. The amplitude decay in damped oscillation is governed by a term involving the damping constant. Reason (R) is false. The angular frequency of a damped oscillation \(omega' = sqrt{omega_0^2 - gamma^2}\) depends on both the natural frequency \(omega_0\) and the damping constant \(gamma\), not solely on \(gamma\).
Assertion (A): General vibrations of a polyatomic molecule about its equilibrium position is periodic but not SHM.
Reason (R): A periodic motion can always be expressed as a sum of infinite number of harmonic motion with appropriated amplitude.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true. Complex vibrations of polyatomic molecules are periodic but generally not simple harmonic motion (SHM). Reason (R) is true. This is the principle of Fourier analysis, stating that any periodic motion can be decomposed into a sum of simple harmonic components. Reason (R) correctly explains why complex periodic motions (like polyatomic vibrations) are not SHM but can still be described as periodic.
Assertion (A): In SHM acceleration leads displacement by phase \(\pi\).
Reason (R): In SHM velocity leads displacement by phase \(\pi/2\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true. If displacement \(x = Asin(\omega t)\), then acceleration \(a = -A\omega^2sin(\omega t) = A\omega^2sin(\omega t + \pi)\). Reason (R) is true. Velocity \(v = A\omega cos(\omega t) = A\omega sin(\omega t + \pi/2)\). Both statements are true, but the phase relationship of velocity with displacement does not explain the phase relationship of acceleration with displacement directly; they are separate facts of SHM.