Question

A body is executing simple harmonic motion. At a displacement $x$, its potential energy is $E_1$ and at a displacement $y$, its potential energy is $E_2$. The potential energy $E$ at a displacement $(x+y)$ is

(Assume that potential energy at mean position is zero)

• A. $E_1+E_2$
• B. $\sqrt{E_1^2+E_2^2}$
• C. $E_1+E_2+2\sqrt{E_1{E}_2}$
• D. $\sqrt{E_1{E}_2}$

Answer 1

The correct option is C $E_1+E_2+2\sqrt{E_1{E}_2}$

Potential energy of a body executing SHM for a displacement $x$ is given by
$E_1=\frac{1}{2}k{x}^2.....\left(1\right)$
Potential energy of a body executing SHM for a displacement $y$ is given by
$E_2=\frac{1}{2}k{y}^2.......\left(2\right)$
Potential energy of a body executing SHM for a displacement $(x+y)$ is given by
$E=\frac{1}{2}k(x+y{)}^2$
$\Rightarrow E=\frac{1}{2}k(x^2+y^2+2xy)$
From $\left(1\right)$ and $\left(2\right)$ we get,
$E=E_1+E_2+k\sqrt{\frac{2E_1}{k}}\sqrt{\frac{2E_2}{k}}$
$\Rightarrow E=E_1+E_2+2\sqrt{E_1{E}_2}$
Thus, option (c) is the correct answer.