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Step-by-Step Solution
Step 1: Recall the velocity expression for Simple Harmonic Motion (SHM)
For a particle executing SHM with angular frequency $ \omega $ and amplitude $ A $, the velocity at a displacement $ x $ from the mean position is given by:
$ v^2 = \omega^2 \bigl(A^2 - x^2\bigr).$
Step 2: Write expressions for the two given conditions
We are given two different positions, $ x_1 $ and $ x_2 $, along with corresponding velocities, $ v_1 $ and $ v_2 $. Applying the velocity expression to each case, we get:
$ v_1^2 = \omega^2 \bigl(A^2 - x_1^2\bigr), \quad v_2^2 = \omega^2 \bigl(A^2 - x_2^2\bigr).
$
Step 3: Eliminate the amplitude $ A $
From the above equations, we see that the amplitude $ A $ can be rewritten in terms of $ x_1 $, $ v_1 $, and $ \omega $. Similarly, it can be written in terms of $ x_2 $, $ v_2 $, and $ \omega $:
$ A^2 = x_1^2 + \frac{v_1^2}{\omega^2} = x_2^2 + \frac{v_2^2}{\omega^2}.
$
Equating the two expressions for $ A^2 $ gives:
$ x_1^2 + \frac{v_1^2}{\omega^2} = x_2^2 + \frac{v_2^2}{\omega^2}.
$
Rearranging terms to solve for $ \omega^2 $:
$ \frac{v_2^2}{\omega^2} - \frac{v_1^2}{\omega^2} = x_1^2 - x_2^2
\quad \Rightarrow \quad
\omega^2 = \frac{v_2^2 - v_1^2}{x_1^2 - x_2^2}.
$
Step 4: Relate angular frequency $ \omega $ to the time period $ T $
For SHM, the time period $ T $ is given by:
$ T = \frac{2\pi}{\omega}.
$
Hence, using $ \omega^2 = \frac{v_2^2 - v_1^2}{x_1^2 - x_2^2} $, we get:
$ \omega = \sqrt{\frac{v_2^2 - v_1^2}{x_1^2 - x_2^2}} \quad \Rightarrow \quad
T = 2\pi \sqrt{\frac{x_1^2 - x_2^2}{v_2^2 - v_1^2}}.
$
Step 5: Rewrite the final expression with correct signs
By convention, we usually write $ (x_2^2 - x_1^2) $ in the numerator and $ (v_1^2 - v_2^2) $ in the denominator to keep the expression positive, depending on which distance is greater from the mean position and which velocity is greater. The final form (as given in the correct answer) is:
$ T = 2\pi \sqrt{\frac{x_2^2 - x_1^2}{v_1^2 - v_2^2}}.
$
This is the desired expression for the time period of the particle’s SHM under the given conditions.
