The work required to stretch a certain spring from an elongation of 4.54 cm to an elongation of 5.54 cm is 31.1 J.


(a) Is the work required to increase the elongation of the spring from 5.54 cm to 6.54 cm greater than, less than, or equal to 31.1 J?

(b) Verify your answer to part (a) by calculating the required work.

a) An already-stretched spring will be harder to pull the second centimeter.

b) The required work is the potential energy change,
(1/2)*k*[2^2 - 1^2] = (3/2) k
You already know that (1/2)k = 31.1 J

To answer part (a) of the question, we can use the concept of elastic potential energy stored in a spring.

The elastic potential energy stored in a spring is given by the equation:

PE = (1/2)kx^2

Where PE is the potential energy, k is the spring constant, and x is the elongation of the spring.

Given that the work required to stretch the spring from an elongation of 4.54 cm to an elongation of 5.54 cm is 31.1 J, we can assume that the potential energy increases by 31.1 J.

So, using the equation for potential energy, we have:

PE = (1/2)kx^2

Since the work done on the spring is equal to the change in potential energy, we have:

31.1 J = (1/2)k(5.54 cm)^2 - (1/2)k(4.54 cm)^2

To determine if the work required to increase the elongation from 5.54 cm to 6.54 cm is greater or smaller than 31.1 J, we need to compare it with the potential energy difference between these two elongations.

The potential energy difference is given by:

ΔPE = (1/2)k(6.54 cm)^2 - (1/2)k(5.54 cm)^2

(b) To verify the answer, we can calculate the required work for the elongation from 5.54 cm to 6.54 cm using the same formula:

ΔPE = (1/2)k(6.54 cm)^2 - (1/2)k(5.54 cm)^2

Then, we compare the calculated work with the work required to stretch the spring from an elongation of 4.54 cm to 5.54 cm.

If the calculated work is greater than 31.1 J, then the work required to increase the elongation is greater than 31.1 J. If it is equal to 31.1 J, then the work is equal. And if it is smaller, then the work is less.

By solving the equation for ΔPE, you can find the answer to part (b) and compare it to 31.1 J to evaluate part (a).