Question

You may use a graphing calculator to solve the following problems. True or False A force is applied to compress a spring several inches. Assume the spring obeys Hooke's Law. Twice as much work is required to compress the spring the second inch than is required to compress the spring the first inch. Justify your answer.

Short answer

False

Step-by-step solution

Understanding Hooke's Law

Hooke's Law states that the force needed to stretch or compress a spring by some distance is proportional to that distance. Mathematically, it is expressed as \( F = kx \), where F is the force applied on the spring, k is the spring constant, and x is the distance the spring stretches or compresses.

Identifying work done

Work done on an object is calculated by the formula \( W = Fd \), where W is the work done, F is the force applied, and d is the distance through which it is moved. However, in the case of a spring, since the force varies as the spring is compressed or stretched, the work done is calculated by integrating the force over the distance. That gives \( W = \frac{1}{2} kx^2 \). The work done to compress a spring is proportional to the square of the distance, not the distance itself.

Applying to the problem

Considering the formula for the work done on a spring, \( W = \frac{1}{2} kx^2 \), the work done to compress the spring the first inch (let's assume x = 1) is \( W = \frac{1}{2} k \). For the second inch (x = 2), the work done is \( W = 2k \). Therefore, it is clear that twice as much work is not required to compress the spring the second inch than is required to compress it the first inch.

Conclusion

Therefore, the statement is False. According to Hooke's Law and the principle of work done, it does not require twice as much work to compress a spring the second inch as compared to the first inch.