1. Write total energy in SHM

For a particle in SHM with amplitude $A$, the total mechanical energy is

$E = \frac{1}{2}kA^2$

where $k$ is the force constant.

2. Write potential energy at displacement $x$

The potential energy at position $x$ is

$PE = \frac{1}{2}kx^2$

3. Write kinetic energy at displacement $x$

Since

$KE = E - PE$

we get

$KE = \frac{1}{2}kA^2 - \frac{1}{2}kx^2 = \frac{1}{2}k(A^2 - x^2)$

4. Use the condition $PE = KE$

Given,

$\frac{1}{2}kx^2 = \frac{1}{2}k(A^2 - x^2)$

Cancel $\frac{1}{2}k$ from both sides:

$x^2 = A^2 - x^2$

$2x^2 = A^2$

$x^2 = \frac{A^2}{2}$

$x = \pm \frac{A}{\sqrt{2}}$

Since the options give only the magnitude of position, the required position is

$\boxed{\frac{A}{\sqrt{2}}}$

5. Check options

• A: $\frac{A}{2}$
• B: $\frac{A}{2\sqrt{2}}$
• C: $\frac{A}{\sqrt{2}}$
• D: $A$

Hence, the correct option is C.