A body is executing simple harmonic motion with frequency \(n\), the frequency of its potential energy is
In SHM, potential energy oscillates with twice the frequency of displacement. Since the displacement frequency is \(n\), the potential energy frequency is \(2n\).
A vertical spring block system is made to oscillate.
Assertion (A): Its time period on earth is more than that on the moon.
Reason (R): Its extension on moon (in equilibrium) is more than that on the earth.
The time period of a vertical spring-block system is \(T = 2\pi \sqrt{\frac{m}{k}}\), which is independent of gravity (g). So (A) is false. The equilibrium extension is \(x_{eq} = \frac{mg}{k}\). Since \(g_{moon} < g_{earth}\), then \(x_{eq,moon} < x_{eq,earth}\). So (R) is also false.
Assertion (A): Total mechanical energy in SHM is conserved.
Reason (R): Kinetic energy of SHM at mean position is equal to potential energy at ends for a particle moving in SHM.
In ideal SHM, total mechanical energy is conserved because the restoring force is conservative. So (A) is true. At the mean position, \(KE_{max} = \frac{1}{2}m(A\omega)^2\), and at the ends, \(PE_{max} = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2A^2\). Thus, \(KE_{mean} = PE_{ends}\). So (R) is true. However, (R) describes a consequence of energy conservation, not the fundamental reason for it.
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.
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 \(-x\).
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.
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.
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).