Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.
b. How much work is required to compress the spring 0.2 m 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 \( F = 15 \) N and \( x = 0.25 \) m, solve for \( k \) by rearranging the formula to \( k = \frac{F}{x} \).
2. Recall that the work \( W \) done in compressing or stretching a spring from the equilibrium position to a displacement \( x \) is given by the integral of the force over the distance, which simplifies to the formula \( W = \frac{1}{2}kx^2 \).
3. Substitute the value of \( k \) found in step 1 and the compression distance \( x = 0.2 \) m into the work formula \( W = \frac{1}{2}kx^2 \).
4. Set up the expression for the work done: \( W = \frac{1}{2} \times k \times (0.2)^2 \).
5. Evaluate the expression to find the amount of work required to compress the spring 0.2 m from its equilibrium position (do not calculate the final numeric value here, just set up the expression).

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 given force and displacement values.

Spring Constant Calculation

The spring constant k measures the stiffness of a spring and is calculated by dividing the applied force by the displacement (k = F/x). Knowing k allows us to analyze forces and work involved in stretching or compressing the spring at different distances.

Work Done on a Spring

The work done to compress or stretch a spring is the energy stored in it, calculated by the integral of force over displacement. For a spring, work W = (1/2)kx², representing the area under the force-displacement curve, which is essential for finding the energy required to compress the spring.