Calculus II/Physics work problem

Question

Given:

30N is required to maintain a spring stretched from its natural length of 12cm to 15cm

 

How much work is done by stretching the string from 16cm to 25cm?

Answer 1

The force required to maintain the spring 3 cm from its natural (non-stretched) length is given by Hooke's law F = kX where k is the spring constant and X is the amount it is stretched (3 cm). The constant k can be calculated by entering 30 N for the and 3 cm for X and solving for k. The gives k= 10 N/cm. As the force is a function of the distance stretched, integration is required to calculate the work required to stretch the spring from 16 to 25 cm. The integration is:

∫ 10N/cm • XdX from 16 to 25 cm giving [10 • X²/2] evaluated from 16 to 25 cm. This gives 3125 -1280 = 1845 joules. As the force required to stretch the spring is in the same direction as the stretch, the work is positive.