Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.
b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position

Verified step by step guidance

1. Recall that the work done to stretch or compress a spring from its equilibrium position is given by the formula \(W = \frac{1}{2} k x^2\), where \(k\) is the spring constant and \(x\) is the displacement from equilibrium.
2. Use the information given for the spring: it requires 2 J of work to stretch it 0.1 m. Substitute these values into the work formula to find the spring constant \(k\): \(2 = \frac{1}{2} k (0.1)^2\).
3. Solve the equation for \(k\) by isolating it: multiply both sides by 2 and divide by \((0.1)^2\) to get \(k = \frac{2 \times 2}{(0.1)^2}\).
4. Now that you have the spring constant \(k\), use it to calculate the work required to stretch the spring 0.4 m by substituting into the work formula: \(W = \frac{1}{2} k (0.4)^2\).
5. Simplify the expression to find the work done for the 0.4 m stretch, which will give you the answer to the problem.

Key Concepts

Hooke's Law

Hooke's Law states that the force required to stretch or compress a spring is proportional to the displacement from its equilibrium position, expressed as F = kx, where k is the spring constant and x is the displacement. This linear relationship is fundamental for calculating forces and work in spring problems.

Work Done by a Variable Force

When stretching a spring, the force varies with displacement, so work is calculated as the integral of force over distance. For springs, work done W = (1/2)kx², representing the area under the force-displacement curve, which is essential for determining energy stored or work required.

Determining the Spring Constant from Work

Given the work done to stretch a spring a certain distance, the spring constant k can be found by rearranging the work formula: k = 2W / x². This allows solving for k when work and displacement are known, enabling calculation of work for other displacements.