Compressing and stretching a spring Suppose a force of 30 N is required to stretch and hold a spring 0.2 m from its equilibrium position.
d. How much additional work is required to stretch the spring 0.2m if it has already been stretched 0.2m from its equilibrium position?

Verified step by step guidance

1. Identify the spring constant \( k \) using Hooke's Law, which states that the force \( F \) required to stretch or compress a spring is proportional to the displacement \( x \) from its equilibrium position: \( F = kx \). Given \( F = 30 \) N and \( x = 0.2 \) m, solve for \( k \) by rearranging the formula to \( k = \frac{F}{x} \).
2. Recall the formula for the work done in stretching or compressing a spring from position \( x = a \) to \( x = b \), which is given by the integral of the force over the displacement: \( W = \int_{a}^{b} kx \, dx \).
3. Set the limits of integration to represent the additional stretch from \( 0.2 \) m to \( 0.4 \) m, since the spring is already stretched 0.2 m and we want the work to stretch it an additional 0.2 m.
4. Evaluate the integral \( W = \int_{0.2}^{0.4} kx \, dx \) by finding the antiderivative \( \frac{kx^2}{2} \) and then computing the difference \( \frac{k(0.4)^2}{2} - \frac{k(0.2)^2}{2} \).
5. Interpret the result as the additional work required to stretch the spring from 0.2 m to 0.4 m beyond its equilibrium position.

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 law helps determine the spring constant from the given force and displacement.

Work Done by a Variable Force

The work done in stretching or compressing a spring is calculated by integrating the force over the displacement, since the force varies with position. For a spring, work W = (1/2) k x^2, representing the energy stored in the spring when stretched or compressed.

Additional Work for Incremental Stretching

To find the additional work required to stretch a spring further from an already stretched position, calculate the difference in work done between the final and initial displacements. This accounts for the extra energy needed beyond the initial stretch.