Question

Assertion (A): In S.H.M kinetic and potential energy become equal when the distance is $\frac{1}{\sqrt{2}}$ times its amplitude. Reason (R): The potential energy of a particle executing S.H.M is periodic with time period being maximum at the extreme displacement.

• (A) and (R) are true. R is correct explanation of A
• (A) and (R) are true. R is not correct explanation of A
• (A) is true, but (R) is false
• (A) is false but (R) is true

Hint:

KE in SHM: $\frac{1}{2}m\omega^2(A^2-x^2)$
PE in SHM: $\frac{1}{2}m\omega^2x^2$ or $\frac{1}{2}kx^2$
Total Energy in SHM: $\frac{1}{2}m\omega^2A^2 = \frac{1}{2}kA^2$ (constant)
Both KE and PE vary periodically, with twice the frequency (half the period) of the oscillation itself.

Answer 1

The Correct Option is B

Assertion (A): In Simple Harmonic Motion (S.H.M.), Kinetic Energy (KE) $= \frac{1}{2}m\omega^2(A^2-x^2)$, where $A$ is amplitude and $x$ is displacement. Potential Energy (PE) $= \frac{1}{2}m\omega^2x^2$. If KE = PE, then $\frac{1}{2}m\omega^2(A^2-x^2) = \frac{1}{2}m\omega^2x^2$. $A^2-x^2 = x^2 \Rightarrow A^2 = 2x^2 \Rightarrow x^2 = \frac{A^2}{2} \Rightarrow x = \pm\frac{A}{\sqrt{2}}$. So, the distance is $\frac{1}{\sqrt{2}}$ times the amplitude. Assertion (A) is true. Reason (R): The potential energy of a particle in S.H.M. is $U(x) = \frac{1}{2}kx^2$. If $x = A \sin(\omega t + \phi)$, then $U(t) = \frac{1}{2}kA^2 \sin^2(\omega t + \phi)$. Since $\sin^2(\theta) = \frac{1-\cos(2\theta)}{2}$, the potential energy varies with an angular frequency of $2\omega$, making it periodic. The potential energy is maximum when $x = \pm A$ (extreme displacement), where $U_{max} = \frac{1}{2}kA^2$. Reason (R) is true. Explanation Check: While both (A) and (R) are true statements, Reason (R) describes properties of potential energy in S.H.M. but does not directly explain why kinetic and potential energies are equal at $x = A/\sqrt{2}$. The equality comes from equating the mathematical expressions for KE and PE, as shown in the analysis of Assertion (A). Therefore, (R) is not the correct explanation of (A). \[ \boxed{\text{(A) and (R) are true. R is not correct explanation of A}} \]