A body executing SHM at a displacement ‘x’ its PE is E1, at a displacement ‘Y’ its PE is E2. The P.E at a displacement (x + y) is

A
$\sqrt{E} = \sqrt{E_{1}} - \sqrt{E_{2}}$
B
$\sqrt{E} = \sqrt{E_{1}} + \sqrt{E_{2}}$
C
E = E₁ + E₂
D
E = E₁ - E₂

Solution:

from definition of potential energy

$E_{1} = \frac{1}{2} k x^{2}; x = \sqrt{\frac{2 E_{1}}{k}} E_{2} = \frac{1}{2} k y^{2}; y = \sqrt{\frac{2 E_{2}}{k}} w h e n d i s p l a c e m e n t i s x + y E = \frac{1}{2} k {(x + y)}^{2} = \frac{1}{2} k {(\sqrt{\frac{2 E_{1}}{k}} + \sqrt{\frac{2 E_{1}}{k}})}^{2} = {(\sqrt{E_{1}} + \sqrt{E_{2}})}^{2} \sqrt{E} = \sqrt{E_{1}} + \sqrt{E_{2}}$