Calculating work for different springs Calculate the work required to stretch the following springs 0.5m from their equilibrium positions. Assume Hooke’s law is obeyed.
a. A spring that requires a force of 50 N to be stretched 0.2 m from its equilibrium position

Verified step by step guidance

1. Identify the given information: the force required to stretch the spring 0.2 m is 50 N, and we want to find the work done to stretch it 0.5 m.
2. Recall Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from equilibrium: \(F = k \times x\), where \(k\) is the spring constant and \(x\) is the displacement.
3. Use the given force and displacement to solve for the spring constant \(k\): rearrange Hooke's Law to \(k = \frac{F}{x}\), then substitute \(F = 50\) N and \(x = 0.2\) m.
4. Calculate the work done to stretch the spring from 0 to 0.5 m using the formula for work done on a spring: \(W = \frac{1}{2} k x^2\), where \(x\) is the final displacement (0.5 m).
5. Substitute the value of \(k\) found in step 3 and \(x = 0.5\) m into the work formula to express the work required to stretch the spring 0.5 m.

Key Concepts

Hooke's Law

Hooke's Law states that the force needed 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 in spring problems.

Spring Constant (k)

The spring constant k measures the stiffness of a spring and is calculated by dividing the force applied by the displacement (k = F/x). Knowing k allows you to determine the force required for any displacement and is essential for computing the work done on the spring.

Work Done on a Spring

The work done to stretch or compress a spring is the energy stored in it, calculated by the integral of force over displacement. For springs obeying Hooke’s Law, work is W = (1/2)kx², representing the area under the force-displacement curve, which is a triangle.