A body is executing a linear S.H.M. Its potential energies at the displacement ' x ' and ' y ' are ' E₁ ' and ' E ₂ ' respectively. Its potential energy at displacement ( x + y ) will be

Step-by-step solution

We know, Potential Energy E _ P = 1 2 Kx ^2 E ₁= 1 2 Kx ^2 x = 2 E ₁ ~K ... (i) and E ₂= 1 2 Ky ^2 y = 2 E ₂ ~K ... (ii) Given, total displacement =(x+y) Potential energy at displacement (x+y) , E is 1 2 ~K ( x + y )^2 aligned & = 1 2 K ( 2 E₁ K + 2 E₂ K )^2 & = 1 2 K [ 2 E₁ K + 2 E₂ K +2 ( E₁ K ) ( 2 E₂ K ) ] & = ( E₁ + E₂ )^2= (E₁+E₂+2 E₁ E₂ ) aligned