The work required to stretch a certain spring from an elongation of 4.73 cm to an elongation of 5.73 cm is 31.7 J.

Question

(a) Is the work required to increase the elongation of the spring from 5.73 cm to 6.73 cm greater than, less than, or equal to 31.7 J?
(b) Verify your answer to part (a) by calculating the required work.

Answer 1

(a) The work required to increase the elongation of the spring from 5.73 cm to 6.73 cm is greater than 31.7 J. This is because the work done on a spring is given by W = (1/2)k(xf^2 - xi^2), where k is the spring constant, and xf and xi are the final and initial elongations, respectively. Since the square of the elongation increases as the elongation increases, the work required to stretch the spring from a larger initial elongation to a larger final elongation is greater than the work required to stretch the spring over a smaller interval.

(b) To calculate the required work, we first need to find the spring constant, k. From the information given, we know that the work required to stretch the spring from an elongation of 4.73 cm to 5.73 cm is 31.7 J. Therefore, we can write:

31.7 J = (1/2)k(5.73^2 - 4.73^2)

Now, we can solve for k:

k = (31.7 J * 2) / (5.73^2 - 4.73^2)
k ≈ 124.28 N/m

Now that we have the spring constant, we can calculate the work required to stretch the spring from an elongation of 5.73 cm to 6.73 cm:

W = (1/2)k(6.73^2 - 5.73^2)
W = (1/2)(124.28 N/m)(6.73^2 - 5.73^2)
W ≈ 34.94 J

Therefore, the work required to increase the elongation of the spring from 5.73 cm to 6.73 cm is about 34.94 J, which is greater than 31.7 J.