Question

A spring with a spring constant of \(92 \mathrm{~N} / \mathrm{m}\) is compressed by \(2.8 \mathrm{~cm}\). How much potential energy is stored in the spring?

Short answer

The potential energy stored in the spring is 0.036064 Joules.

Step-by-step solution

Identify the Variables

First, we need to identify the variables in the problem. The spring constant is given as \( k = 92 \text{ N/m} \) and the compression distance is \( x = 2.8 \text{ cm} \). Since the potential energy formula requires the distance in meters, convert \( x \) to meters: \( x = 2.8 \text{ cm} = 0.028 \text{ m} \).

Write the Formula for Spring Potential Energy

The potential energy stored in a spring is calculated using the formula: \[ PE = \frac{1}{2} k x^2 \] where \( PE \) is potential energy, \( k \) is the spring constant, and \( x \) is the compression or elongation of the spring in meters.

Substitute the Values Into the Formula

Substitute the given values into the potential energy formula: \[ PE = \frac{1}{2} \times 92 \times (0.028)^2 \] Performing these calculations will give us the potential energy stored in the spring.

Perform the Calculations

First, calculate the square of the compression distance: \( (0.028)^2 = 0.000784 \).Next, multiply this by the spring constant and \( \frac{1}{2} \): \[ PE = \frac{1}{2} \times 92 \times 0.000784 = 0.036064 \text{ Joules} \].

Key concepts

Spring Constant

The spring constant, often denoted by the letter \( k \), is a measure of a spring's stiffness. A larger spring constant means a stiffer spring, which requires more force to compress or stretch it by a certain distance.
Conversely, a smaller spring constant indicates a more flexible spring. The spring constant is measured in newtons per meter (N/m).
These units show the amount of force needed to change the spring's length by one meter.

• Understanding through Units: If a spring has a high spring constant, even a small compression will require a significant force.
• Real-World Applications: Springs with different constants are used in applications like car suspensions, mattresses, and watches. In each of these, the appropriate spring constant is chosen to provide desired comfort or functionality.

Hooke's Law

Hooke's Law is a core principle when studying springs. It describes how the force exerted by a spring is proportionate to its change in length. This can be expressed with the equation \( F = -kx \), where
\( F \) is the force applied to the spring, \( k \) is the spring constant, and \( x \) is the displacement from the spring's original length.

• Direct Relationship: The minus sign shows that the direction of the force is opposite to the direction of displacement.
• Proportionality: If you double the displacement \( x \), the force \( F \) will also double, assuming the spring remains within its elastic limit.
• Elastic Limit: Beyond a certain point, known as the elastic limit, Hooke's Law no longer applies, and the spring may permanently deform.

Energy Conversion

In this exercise, we see energy conversion from mechanical work to potential energy within the spring. When you compress a spring, the work you do on it is stored as potential energy. This energy can later be converted back into kinetic energy when the spring is released.

• Potential Energy Formula: The potential energy (PE) stored in a spring is given by \[ PE = \frac{1}{2} k x^2 \]where \( k \) is the spring constant and \( x \) is the change in length.
• Energy Transformation: This stored energy can do work on other objects or be released as kinetic energy, depending on the setup.
• Practical Example: In a toy gun, a spring is compressed to load it. Upon release, the spring's potential energy propels a projectile forward.