Question

If $x$, $v$ and $a$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then which of the following do not change with time?

• A. $aT/v$
• B. $aT + 2\pi v$
• C. $a^2T^2 + 4\pi^2 v^2$
• D. $aT$
• E. $vT$

Hint:

Any expression in SHM that simplifies to a function of only the amplitude ($A$) and frequency ($\omega$) will be time-independent.

Answer 1

The Correct Option is C

Concept:
In Simple Harmonic Motion (SHM), we use the following relationships: [itemsep=6pt]
• $a = -\omega^2 x$
• $v^2 = \omega^2(A^2 - x^2) \implies v^2 + \omega^2 x^2 = \omega^2 A^2$
• $T = 2\pi / \omega \implies \omega T = 2\pi$

Step 1: Substituting variables into Option (C).
Consider the expression $a^2T^2 + 4\pi^2 v^2$: \[ (-\omega^2 x)^2 T^2 + 4\pi^2 v^2 = \omega^4 x^2 T^2 + 4\pi^2 v^2 \] Since $\omega T = 2\pi$, we can substitute $\omega^2 T^2 = 4\pi^2$: \[ \omega^2 x^2 (4\pi^2) + 4\pi^2 v^2 = 4\pi^2 (\omega^2 x^2 + v^2) \]

Step 2: Evaluating the result.
From the SHM velocity equation, we know $\omega^2 x^2 + v^2 = \omega^2 A^2$. \[ 4\pi^2 (\omega^2 A^2) \] Because $4, \pi, \omega,$ and $A$ are all constants, the entire expression remains constant over time.