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 \(2n\).
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): 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): We can assume damped oscillation to be approximately periodic motion for small damping
Reason (R): Small damping means \( \frac{b}{\sqrt{km}} \ll 1 \)
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): 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 \(gamma\).
Assertion (A): In damped oscillation both amplitude and frequency change with time.
Reason (R): Both amplitude and frequency vary exponentially.
In a damped oscillation, the amplitude decreases exponentially with time, but the frequency of oscillation (for underdamped case) remains constant. Therefore, both Assertion (A) and Reason (R) are false.
Assertion (A): In simple harmonic motion total mechanical energy can be negative also.
Reason (R): Potential energy is always negative and if it is greater than kinetic energy total mechanical energy will be negative.
In simple harmonic motion, the total mechanical energy is \(E = frac{1}{2} kA^2\). Since the spring constant \(k\) and amplitude \(A\) are real and positive, the total energy \(E\) is always positive. Potential energy in SHM, \(U = frac{1}{2} kx^2\), is always positive or zero. Thus, both Assertion (A) and Reason (R) are false.
Assertion (A): The graph of potential energy and kinetic energy of a particle in SHM with respect to position is a parabola.
Reason (R): The potential energy and kinetic energy of a particle in SHM, do not vary linearly with position.
Potential energy in SHM is \( PE = \frac{1}{2} kx^2 \), which is a parabola. Kinetic energy is \( KE = \frac{1}{2} k(A^2 - x^2) \), also a parabola. So (A) is true. Since both are quadratic functions of position \( x \), they do not vary linearly. Thus, (R) is true and correctly explains (A).