Question

A force of \(27 \mathrm{~N}\) stretches a given spring by \(4.4 \mathrm{~cm}\). How much potential energy is stored in the spring when it is compressed \(3.5 \mathrm{~cm}\) ?

Short answer

The potential energy stored is approximately 0.375 Joules.

Step-by-step solution

Understand Hooke's Law

Hooke's Law states that the force needed to extend or compress a spring is proportional to the distance it is stretched or compressed. Mathematically, this is expressed as \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.

Calculate the Spring Constant

We know that a force of 27 N stretches the spring by 4.4 cm. First, convert the displacement from cm to meters: \( 4.4 \text{ cm} = 0.044 \text{ m} \). Using Hooke's Law, set up the equation \( 27 = k \times 0.044 \). Solve for \( k \):\[k = \frac{27}{0.044} = 613.64 \ ext{N/m}.\]

Use the Formula for Elastic Potential Energy

The elastic potential energy stored in a spring is given by the formula \( U = \frac{1}{2} k x^2 \), where \( U \) is the potential energy, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.

Calculate Potential Energy for Compression

The spring is compressed by 3.5 cm, which needs to be converted to meters: \( 3.5 \text{ cm} = 0.035 \text{ m} \). Substitute the known values into the formula for potential energy:\[U = \frac{1}{2} \times 613.64 \times (0.035)^2.\]Calculate the energy:\[U = \frac{1}{2} \times 613.64 \times 0.001225 = 0.375 \ ext{Joules}.\]

Conclusion of the Calculation

The potential energy stored in the spring when it is compressed by 3.5 cm is approximately 0.375 Joules.

Key concepts

Spring Constant

The spring constant, often symbolized as "\( k \)," is a measure of a spring's stiffness. It tells us how much force is needed to stretch or compress the spring by a certain distance. This concept is at the heart of Hooke's Law.

You can think of the spring constant as how tough or easy-going a spring is. A larger spring constant means that the spring is stiff and requires more force to change its length. Conversely, a smaller constant means the spring is more pliable and easier to stretch.

In our example, we calculated the spring constant by using the relationship given by Hooke's Law, \( F = kx \). Given a force of 27 N and a displacement of 0.044 m (converted from 4.4 cm), we found the spring constant to be \( k = 613.64 \, \mathrm{N/m} \).

• This calculation demonstrates that for every meter the spring is stretched, 613.64 N of force is needed.
• Understanding the spring constant helps in determining other properties of the spring, like potential energy.

Elastic Potential Energy

Elastic potential energy is the energy stored in elastic materials, such as springs, when they are compressed or stretched. It's a type of potential energy and is important for understanding how the energy can be retrieved from a spring.

The formula for elastic potential energy is \( U = \frac{1}{2} k x^2 \), where \( U \) represents the potential energy, \( k \) is the spring constant, and \( x \) is the distance from the equilibrium position. This equation shows that elastic potential energy increases with the square of the displacement, meaning small changes in \( x \) can greatly change \( U \).

In our case, we calculated the elastic potential energy when the spring was compressed by \( 3.5 \text{ cm} \). The potential energy turned out to be approximately \( 0.375 \text{ Joules} \).

• This indicates just how much energy is "stored" in the spring due to its compression.
• This stored energy can be released to do work, like pushing or pulling objects attached to the spring.

Compression and Stretching of Springs

The concepts of compression and stretching are fundamental to spring dynamics. When a spring is compressed or stretched, it moves from its natural equilibrium position, requiring or releasing energy.

Compression refers to reducing the length of the spring, while stretching refers to increasing its length. Both actions require applying force; the amount of force depends on the spring constant and how much the spring is compressed or stretched.

Applying Hooke's Law gives us insight into the relationship of force and displacement \( x \). The

• The greater the distance a spring is compressed or stretched, the more force is needed.
• Similarly, the potential energy stored in the spring is proportional to the square of the displacement \((x^2)\).

This non-linear relationship means that even a small change in the spring’s compression or stretching can significantly alter the potential energy stored in the spring.

Understanding these principles is crucial when calculating how much energy a spring can store and how it can be applied in practical scenarios, like in shock absorbers or trampolines.