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

• 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

$E_1+E_2+2\sqrt{E_1{E}_2}$

$E_1=\frac{1}{2}m{\omega }^2{x}^2$
$\Rightarrow \sqrt{E_1}=x\sqrt{\frac{1}{2}m{\omega }^2}$ ..... (1)
Similarly, $E_2=\frac{1}{2}m{\omega }^2{y}^2$
$\Rightarrow \sqrt{E_2}=y\sqrt{\frac{1}{2}m{\omega }^2}$ ..... (2)
Now, $E=\frac{1}{2}m{\omega }^2(x+y{)}^2$
$\Rightarrow \sqrt{E}=(x+y)\sqrt{\frac{1}{2}m{\omega }^2}$ ..... (3)
From (1), (2) and (3), it follows that
$\sqrt{E}=\sqrt{E_1}+\sqrt{E_2}$
$\Rightarrow E=E_1+E_2+2\sqrt{E_1{E}_2}$
Hence, the correct choice is (c).