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.
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.
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.
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): Under forced oscillation external periodic force apply to sustain the motion.
Reason (R): Under forced oscillation phase of harmonic motion of the particle differs from the phase of the driving force.
Assertion (A) is true: Forced oscillations involve an external periodic force to maintain motion. Reason (R) is true: In forced oscillations, a phase difference exists between the driving force and the particle's motion. (R) describes a property of forced oscillations, but doesn't explain the reason for applying the external force as stated in (A).
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.
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): \(x = \sin^2(\omega t)\) represents a SHM about mean position \(x = \frac{1}{2}\).
Reason (R): \(a \propto -x\) is the necessary condition for SHM.
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.
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.
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.
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.
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 \(gamma\).