Question

 A body is executing $SHM$, At a displacement $x$, its potential energy is ${E_1}$​ and at a displacement $y$, its potential energy is ${E_2}$​. Find its potential energy $E$ at displacement $(x + y)$
A. $\sqrt E = \sqrt {{E_1}} - \sqrt {{E_2}} $
B. $\sqrt E = \sqrt {{E_1}} + \sqrt {{E_2}} $
C. None of these
D. All of these

Answer 1

Hint:
A simple harmonic motion is one in which the restoring force constantly goes in the direction of the mean position and is directly proportional to the body's deviation from its mean position. To solve this question we need to understand the relationship between potential energy and displacement of the particle doing SHM.

Formula used:
${E} = \dfrac{1}{2}k{x^2}\,\,\,...(i)$



Complete step by step solution:

In order to know that according to the definition of potential energy, "It is the energy that depends on the relative placement of the various components of the system." Potential Energy develops in systems with pieces whose configuration or relative positions cause forces on each other that vary in magnitude.
In short, potential energy is the power that an object can store due to its position in relation to other things, internal tensions, electric charge, or other circumstances.
For a displacement $x$, the potential energy of a body performing simple harmonic motion is given by:
${E_1} = \dfrac{1}{2}k{x^2}\,\,\,...(i)$
Similarly, for a displacement$y$, the potential energy of a body performing simple harmonic motion is given by:
${E_2} = \dfrac{1}{2}k{y^2}\,\,\,...(ii)$
Now, for a displacement$(x + y)$, the potential energy of a body performing simple harmonic motion is given by:
$E = \dfrac{1}{2}k{(x + y)^2} \\$
 $\Rightarrow E = \dfrac{1}{2}k({x^2} + {y^2} + 2xy) \\$
To determine the its potential energy $E$ at displacement $(x + y)$, from $(i)$and $(ii)$, then we obtain:
$E = {E_1} + {E_2} + k\sqrt {\dfrac{{2{E_1}}}{k}} \sqrt {\dfrac{{2{E_2}}}{k}} \\$
 $\Rightarrow E = {E_1} + {E_2} + 2\sqrt {{E_1}{E_2}} \\$
  $\Rightarrow \sqrt E = \sqrt {{E_1}} + \sqrt {{E_2}} \\$
Thus, the correct option is: (B) $\sqrt E = \sqrt {{E_1}} + \sqrt {{E_2}} $



Note:
It should be noted that to solve this problem, we must thoroughly analyse it and use the appropriate formulas. In addition to applying the proper formula to determine the potential energy at displacement $(x + y)$.