Hooke's Law In Exercises 9-14' use Hooke's Law to determine...

Hooke's Law In Exercises 9-14' use Hooke's Law to determine the work done by the variable force in the spring problem.

A force of 5 pounds compresses a 15-inch spring a total of 3 inches. How much work is done in compressing the spring 7 inches?

Key Concepts

Hooke's Law

Hooke's Law is a principle of physics that states that, for relatively small deformations, the force exerted by a spring is directly proportional to the displacement from its equilibrium position. This relationship is commonly expressed as F = kx, where F is the force applied, x is the displacement, and k is the spring constant.

Spring Constant

The spring constant, denoted by k, quantifies the stiffness of a spring. It is determined by the ratio of the force applied to the displacement produced (k = F/x). A higher spring constant indicates a stiffer spring that requires more force to produce the same displacement as a less stiff spring.

Work Done by a Variable Force

When the force applied is not constant but varies with displacement, the work done is calculated by integrating the force over the distance moved. In the context of a spring following Hooke’s Law, the force is variable (F = kx), so the work done in compressing or stretching the spring from one displacement to another is found by evaluating the integral of kx with respect to x.

Definite Integration

Definite integration is a mathematical tool used to calculate the accumulated value of a function over an interval. In this context, it is used to sum up all the infinitesimal amounts of work done by the variable force over the displacement of the spring, leading to the formula for work as the integral of kx dx from the initial to the final displacement.

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