Potential Energy Stored in a Spring

Definition: The spring constant, k , is a measure of the stiffness of a spring (large k $\rightarrow$ stiff spring, small k $\rightarrow$ soft spring).

To compress a spring by a distance $\Delta$x we must apply a force F _ext = k$\Delta$x . By Newton's 3rd law, if we hold a spring in a compressed position, the spring exerts a force F_s = - k$\Delta$x . This is called a linear restoring force because the force is always in the opposite direction from the displacement.

Note:

• The sign of F_s shows that the spring resists attempts to compress or stretch it; therefore F_s is a restoring force.

  For Example: In Figure (5.2a) $\Delta$x = x_f - x_i = - 5 which gives F_s = - k(- 5) = 5k . This force is positive and therefore directed to the right. This means that the spring resists the compression. In Figure (5.2b) $\Delta$x = x_f - x_i = 3 which gives F_s = - 3k . The negative sign indicates that the force is to the left and that the spring resists the stretching.
  
  

   

  Figure: a) Compressed spring b) Stretched spring

  
  
  
• The farther we compress or stretch the spring, the greater the restoring force.
• We usually define x_i = 0 and x_f = x which gives F_s = - kx . This is called Hooke's law.

To find the potential energy stored in a compressed (or stretched) spring, we calculate the work to compress (or stretch) the spring: the force to compress a spring varies from F _ext = F₀ = 0 (at x_i = 0 ), to F _ext = F_x = kx (at x_f = x ). Since force increases linearly with x , the average force that must be applied is

$$\bar{F}_{ext}^{} {\textstyle\frac{1}{2}} {\textstyle\frac{1}{2}}$$

The work done by $\bar{F}_{ext}^{}$ is W = $\bar{F}_{ext}^{}$x = ${\frac{1}{2}}$kx ². This work is stored in the spring as potential energy:

$${\textstyle\frac{1}{2}}$$ (4)

Note:

• PE_s = 0 when x = 0 (at equilibrium).
• PE_s always > 0 when the spring is not in equilibrium.
• PE_s is the same if x = $\pm$ x_f (same PE_s for equal expansion or compression).