A body is executing simple harmonic motion. At a displacement x, its poten

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)

Options:

A . E1+E2

B . √E21+E22

C . E1+E2+2√E1E2

D . √E1E2

Answer: Option C
:
C

E_{1} = \frac{1}{2} m ω^{2} x^{2} o r \sqrt{E_{1}} = x \sqrt{\frac{1}{2} m ω^{2}} . . . . . . . (1)

E_{2} = \frac{1}{2} m ω^{2} y^{2} o r \sqrt{E_{2}} = y \sqrt{\frac{1}{2} m ω^{2}} . . . . . . . (2)

E = \frac{1}{2} m ω^{2} (x + y)^{2} o r \sqrt{E} = (x + y) \sqrt{\frac{1}{2} m ω^{2}} . . . . . . . (3)

From (1), (2) and (3) it follows that

\sqrt{E} = \sqrt{E_{1}} + \sqrt{E_{2}}

O r E = E_{1} + E_{2} + 2 \sqrt{E_{1} E_{2}}

Which is choice (c).

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