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. $\sqrt{E_1{E}_2}$
• D. $E_1+E_2+2\sqrt{E_1{E}_2}$

Answer 1

Answer: (A), (C)

The correct options are
A $E_1+E_2$
C $\sqrt{E_1{E}_2}$

At displacement x

$E_1=\frac{1}{2}m{\omega }^2{x}^2$

or $\sqrt{E_1}=x\sqrt{\frac{1}{2}m{\omega }^2}$ ...(i)

At displacement y

$E_2=\frac{1}{2}m{\omega }^2(x+y{)}^2$

or $\sqrt{E_2}=(x+y)\sqrt{\frac{1}{2}m{\omega }^2}$ ...(ii)

At displacement (x+y)

$E=\frac{1}{2}m{\omega }^2(x+y{)}^2$

or $\sqrt{E}=(x+y)\sqrt{\frac{1}{2}m{\omega }^2}$ ...(iii)

From Eqs. (i), (ii) and (iii) it follows that

$\sqrt{E}=\sqrt{E_1}+\sqrt{E_2}$

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