Suppose that 4 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 47 cm.(a) How much work is needed to stretch the spring from 38 cm to 43 cm? (Round your answer to two decimal places.) J(b) How far beyond its natural length will a force of 20 N keep the spring stretched? (Round your answer one decimal place.) cm
Question
Suppose that 4 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 47 cm.(a) How much work is needed to stretch the spring from 38 cm to 43 cm? (Round your answer to two decimal places.) J(b) How far beyond its natural length will a force of 20 N keep the spring stretched? (Round your answer one decimal place.) cm
Solution
The problem involves Hooke's Law, which states that the force required to stretch or compress a spring by some distance x is proportional to x. The work done on the spring is the integral of this force over the distance the spring is stretched or compressed.
(a) We know that 4 J of work is needed to stretch the spring from 30 cm to 47 cm, a distance of 17 cm. The work done per unit distance is therefore 4 J / 17 cm = 0.235 J/cm. To find the work needed to stretch the spring from 38 cm to 43 cm, a distance of 5 cm, we multiply this distance by the work done per unit distance: 5 cm * 0.235 J/cm = 1.18 J.
(b) The force required to stretch the spring is proportional to the distance it is stretched. We know that 4 J of work is needed to stretch the spring 17 cm, so the force required is 4 J / 17 cm = 0.235 N/cm. If a force of 20 N is applied to the spring, it will be stretched by a distance of 20 N / 0.235 N/cm = 85.1 cm beyond its natural length.
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