Question

Find the associated potential energy for the force exerted by an elastic spring.

Show the given, required, equation, solution and answer.

Answer 1

Explanations & Calculations

Given:

• The force ($\small F$ ), exerted by the spring.
• The elastic coefficient ($\small k$ ) of the spring.

Required:

• The Hook's law equation.
• The amount of extended length ($\small x$ ).
• Concept of work done by a force.

Equation:

$\qquad\qquad \begin{aligned} \small W&=\small Fs\\ \small F&=\small kx \end{aligned}$

Solution:

• Work done by the applied force. The force, in the beginning, is zero and $\small F$ at the required point.
• Therefore, the median force needs to be considered in finding the work done.
• The work done on the spring to extend it to the required point is stored in it as elastic potential energy.
• Therefore,

$\qquad\qquad \begin{aligned} \small F_{median}&=\small \frac{0+F}{2}=\frac{F}{2}\\ \\ \small W&=\small E_p= \frac{Fx}{2}\cdots\cdots(1)\\ \\ \small F&=\small kx\cdots\cdots(2)\\ \\ &\small (2) \to (1) \\ \\ \small E_p&=\small \frac{F^2}{2k} \end{aligned}$

Answer:

$\qquad\qquad\qquad\qquad\small E_p = \large\frac{F^2}{2k}$