# Potential Energy Stored in a Spring

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

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

Note:

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

For Example: In Figure (5.2a) x = xf - xi = - 5 which gives Fs = - 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) x = xf - xi = 3 which gives Fs = - 3k . The negative sign indicates that the force is to the left and that the spring resists the stretching.

• The farther we compress or stretch the spring, the greater the restoring force.
• We usually define xi = 0 and xf = x which gives Fs = - 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 xi = 0 ), to F ext = Fx = kx (at xf = x ). Since force increases linearly with x , the average force that must be applied is

 = (F0 + Fx) = kx
The work done by is W = x = kx 2. This work is stored in the spring as potential energy:
 PEs = kx 2. (4)
Note:
• PEs = 0 when x = 0 (at equilibrium).
• PEs always > 0 when the spring is not in equilibrium.
• PEs is the same if x = xf (same PEs for equal expansion or compression).

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