Question

A system is executing S.H.M. The potential energy of the system for displacement x is E₁ and for a displacement of y, the potential energy of the system is E₂. The potential energy for a displacement of (x + j) is……..

(A) E₁ + E₂

(B) √(E₁² + E₂²)

(C) E₁ + E₂ + 2√E₁E₂

(D) √E₁E₂

Answer 1

The correct option (C) E₁ + E₂ + 2 √E₁E₂

Explanation:

PE = (1/2)mω²x² ---- general formula

1^st case.

E₁ = (1/2) mω²x²  (1)

2^nd case

E₂     = (1/2) mω²y²  (2)

let PE for the displacement (x + y) is E

∴ E  = (1/2) mω² (x + y)²

= (1/2) mω²(x² + 2xy + y²)

= (1/2) mω² x² + (1/2) mω²y² + mω² xy

from (1) & (2)

E = E₁ + E₂ + mω² xy  (3)

also  E₁ × E₂ = {(1/2) mω²}² x² y²

i.e. 4E₁ E₂  = (mω² xy)²

Taking square roots,

mω² xy = √(4 E₁E₂)

hence from (3)

E = E₁ + E₂ + 2 √E₁E₂