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).
Assertion (A): Maximum potential energy in simple harmonic motion is equal to net mechanical energy.
Reason (R): Maximum kinetic energy in simple harmonic motion is equal to net mechanical energy.
In SHM, total mechanical energy \( E \) is conserved. At extreme positions (maximum displacement), kinetic energy is zero, so \( E = PE_{max} \). Thus (A) is true. At the equilibrium position, potential energy is zero, so \( E = KE_{max} \). Thus (R) is true. However, (R) does not explain (A); both are independent statements describing energy distribution in SHM.
Assertion (A): If the amplitude of a simple harmonic oscillator is doubled, its total energy also becomes doubled.
Reason (R): In harmonic oscillation, the total energy is directly proportional to the amplitude of vibration.
The total energy of an SHM is `\(E = \frac{1}{2}kA^2\)`. If amplitude `\(A\)` is doubled, energy becomes `\(E' = \frac{1}{2}k(2A)^2 = 4E\)`. So A is false.
Reason R states energy is directly proportional to amplitude, which is also false (it's proportional to `\(A^2\)`). Both are false.
Assertion (A): For a system executing SHM, the mechanical energy remains constant.
Reason (R): In SHM, kinetic energy and potential energy vary periodically with double the frequency of SHM.
For an ideal SHM, mechanical energy is conserved (A is true). Kinetic energy `\(KE = \frac{1}{2}m\omega^2A^2\cos^2(\omega t)\)` and potential energy `\(PE = \frac{1}{2}k A^2\sin^2(\omega t)\)` vary with `\(2\omega\)`, double the SHM frequency (R is true).
However, R describes the variation of KE/PE, not the reason for conservation of total mechanical energy. Thus, R does not explain A.
A particle is undergoing SHM having total mechanical energy equal to \(8\text{ J}\). At an instant its kinetic energy is found to be \(10\text{ J}\), then its potential energy at that instant is
Total mechanical energy \(E\) in SHM is the sum of kinetic energy \(K\) and potential energy \(U\): \(E = K + U\). Substituting the values, \(8\text{ J} = 10\text{ J} + U\), which gives \(U = -2\text{ J}\) as the potential energy at that instant.
A simple pendulum hanging freely stayed at rest in vertical, because in this position
A stable equilibrium state corresponds to a local minimum of the system's potential energy. For a simple pendulum, the lowest point is the vertical position, where potential energy is minimum.
For the damped oscillator, if time taken for its amplitude of vibrations to drop to half of its initial value is \(T\) then time taken for amplitude to drop to one eighth amplitude is
Amplitude decays exponentially as \(A = A_0 e^{-\gamma t}\). Since it halves in time \(T\), to drop to \(1/8 = (1/2)^3\) of its initial value, it takes exactly \(3T\).