Assertion (A): A SHM may be assumed as composition of many SHM’s.
Reason (R): Superposition of many SHM’s (along same line) of same frequency will be a SHM.
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) states that an SHM can be viewed as a composition of multiple SHMs. Reason (R) states that the superposition of multiple SHMs along the same line and with the same frequency results in another SHM.
Both statements are true, and (R) provides the explanation for how (A) can be possible.
Assertion (A): Displacement-time equation of a particle moving along \(x\)-axis is \(x = 4 + 6 sin\omega t\). Under this situation, motion of particle is not simple harmonic.
Reason (R): \(\frac{d^2x}{dt^2}\) for the given equation is not proportional to \(-x\).
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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For SHM, \(a = -\omega^2 (x-x_0)\). Given \(x = 4 + 6 sin\omega t\), the equilibrium position is \(x_0=4\). The acceleration is \(\frac{d^2x}{dt^2} = -\omega^2 (x-4)\). If SHM is strictly defined as \(a \propto -x\) (equilibrium at origin), then (A) is true. (R) is also true as \(\frac{d^2x}{dt^2}\) is proportional to \(-(x-4)\), not \(-x\). (R) explains (A).
Assertion (A): For a particle performing SHM, its speed decreases as it goes away from the mean position.
Reason (R): In SHM, the acceleration is always opposite to the velocity of the particle.
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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As a particle in SHM moves from the mean to extreme position, its speed decreases as the restoring force opposes motion. So (A) is true. Acceleration is always directed towards the equilibrium. When moving towards equilibrium, velocity and acceleration are in the same direction, so (R) is false.
Assertion (A): Motion of a ball bouncing elastically in vertical direction on a smooth horizontal floor is a periodic motion but not an SHM.
Reason (R): Motion is SHM when restoring force is proportional to displacement from mean position.
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
View Answer
Periodic motion repeats over time. SHM requires a restoring force proportional to displacement \(F = -kx\). A bouncing ball experiences gravitational force \(F=mg\) (constant) and impulsive forces upon impact, not a linear restoring force. Hence, it's periodic but not SHM, and the reason correctly defines SHM.
Assertion (A): A particle, simultaneously subjected to two simple harmonic motions of same frequency and same amplitude, will perform SHM only if the two SHM’s are in the same direction.
Reason (R): A particle, simultaneously subjected to two simple harmonic motions of same frequency and same amplitude, perpendicular to each other the particle can be in uniform circular motion.
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: Superposition of parallel SHMs results in SHM. Assertion (R) is true: Perpendicular SHMs of same frequency and amplitude with \(frac{pi}{2}\) phase difference result in UCM. However, (R) explains a scenario where SHM is not formed along a line, not why (A) is true.
Assertion (A): \(x = A sin \omega t\) & \(y = B cos \omega t\) In the above co-ordinates particle moves in elliptical path.
Reason (R): A periodic motion can always be expressed as a sum of infinite number of harmonic motions with appropriate 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: From \(x = A sin \omega t\) and \(y = B cos \omega t\), we get \(\left( \frac{x}{A} \right)^2 + \left( \frac{y}{B} \right)^2 = 1\), which is an ellipse. Reason (R) is true: Fourier's theorem states any periodic motion can be decomposed into harmonic components. However, (R) does not explain why the given equations describe an ellipse.
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): 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.