Six and one-half foot-pounds of work is required to compress a spring 3 inches from its natural length. Find the work required to compress the spring an additional one-half inch.
To find the work required to compress the spring an additional one-half inch, we can use the formula for work done on a spring:
Work = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the natural length.
Given that 6 and one-half foot-pounds of work is required to compress the spring 3 inches, we can calculate the spring constant, k:
Work = (1/2) * k * (3 inches)^2
6.5 foot-pounds = (1/2) * k * 9 inches^2
k = (6.5 foot-pounds * 2) / (9 inches^2)
k = (13 foot-pounds) / (81 inches^2)
Now, we can find the work required to compress the spring an additional one-half inch:
x = 3 inches + 0.5 inches = 3.5 inches
Work = (1/2) * k * (3.5 inches)^2
Work = (1/2) * [(13 foot-pounds) / (81 inches^2)] * (3.5 inches)^2
Work = (1/2) * (13 foot-pounds) / (81 inches) * (3.5 inches)^2
Work = (1/2) * (13 foot-pounds) * (3.5 inches)^2 / (81 inches)
Work ≈ 0.0804 foot-pounds
Therefore, approximately 0.0804 foot-pounds of work is required to compress the spring an additional one-half inch.