EXAMPLE 8 3 Spring Potential Energy A system contains a perfectly elastic spring, with an unstretched length of 20 cm and a spring constant of 4 N cm (a) How much elastic potential energy does the spring contribute when its length is 23 cm (b) How much more potential energy does it contribute if its length increases to 26 cm Strategy When the spring is at its unstretched length, it contributes nothing to the potential energy of the system, so we can use Equation 8 7 with the constant equal to zero The value of x is the length minus the unstretched length When the spring is expanded, the spring's displacement or difference between its relaxed length and stretched length should be used for the x value in calculating the potential energy of the spring Solution The displacement of the spring is x 23cm20cm 3cm 23cm20cm 3cm, so the contributed potential energy is U 12kx2 12(4N cm)(3cm)2 0 18J 122 12(4N cm)(3cm)2 0 18J When the spring's displacement is x 26cm20cm 6cm 26cm20cm 6cm, the potential energy is U 12kx2 12(4N cm)(6cm)2 0 72J 122 12(4N cm)(6cm)2 0 72J, which is a 0 54 J increase over the amount in part (a) Significance Calculating the elastic potential energy and potential energy differences from Equation 8 7 involves solving for the potential energies based on the given lengths of the spring Since U depends on x22, the potential energy for a compression (negative x ) is the same as for an extension of equal magnitude CHECK YOUR UNDERSTANDING 8 3 When the length of the spring in Example 8 3 changes from an initial value of 22 0 cm to a final value, the elastic potential energy it contributes changes by 0 0800J 0 0800J Find the final length

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EXAMPLE 8.3 Spring Potential Energy A system contains a perfectly elastic spring, with an unstretched length of 20 cm and a spring constant of 4

EXAMPLE 8.3

Spring Potential Energy

A system contains a perfectly elastic spring, with an unstretched length of 20 cm and a spring constant of 4 N/cm. (a) How much elastic potential energy does the spring contribute when its length is 23 cm? (b) How much more potential energy does it contribute if its length increases to 26 cm?

Strategy

When the spring is at its unstretched length, it contributes nothing to the potential energy of the system, so we can use Equation 8.7 with the constant equal to zero. The value of x is the length minus the unstretched length. When the spring is expanded, the spring's displacement or difference between its relaxed length and stretched length should be used for the x-value in calculating the potential energy of the spring.

Solution

The displacement of the spring is x=23cm20cm=3cm=23cm20cm=3cm, so the contributed potential energy is U=12kx2=12(4N/cm)(3cm)2=0.18J=122=12(4N/cm)(3cm)2=0.18J.
When the spring's displacement is x=26cm20cm=6cm=26cm20cm=6cm, the potential energy is U=12kx2=12(4N/cm)(6cm)2=0.72J=122=12(4N/cm)(6cm)2=0.72J, which is a 0.54-J increase over the amount in part (a).

Significance

Calculating the elastic potential energy and potential energy differences from Equation 8.7 involves solving for the potential energies based on the given lengths of the spring. Since U depends on x22, the potential energy for a compression (negative x) is the same as for an extension of equal magnitude.

CHECK YOUR UNDERSTANDING 8.3

When the length of the spring in Example 8.3 changes from an initial value of 22.0 cm to a final value, the elastic potential energy it contributes changes by 0.0800J.0.0800J. Find the final length.