Question

It takes \(13 \mathrm{~N}\) to stretch a certain spring \(9.5 \mathrm{~cm}\). How much potential energy is stored in this spring? (Hint: Calculate the spring constant first, then the potential energy.)

Short answer

The potential energy stored in the spring is \(0.617 \mathrm{~J}\).

Step-by-step solution

Find the spring constant

Use Hooke's Law, which states \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement. Rearrange the formula to find \( k \): \( k = \frac{F}{x} \). Plug in the given values: \( F = 13 \mathrm{~N} \) and \( x = 9.5 \mathrm{~cm} = 0.095 \mathrm{~m} \). So, \( k = \frac{13}{0.095} = 136.84 \mathrm{~N/m} \).

Calculate the potential energy stored in the spring

Use the formula for potential energy stored in a spring: \( PE = \frac{1}{2}kx^2 \). Substitute the spring constant \( k = 136.84 \mathrm{~N/m} \) and the displacement \( x = 0.095 \mathrm{~m} \) into the formula. So, \( PE = \frac{1}{2} \times 136.84 \times (0.095)^2 = 0.617 \mathrm{~J} \).

Key concepts

Spring Constant

In the world of springs, the spring constant, often represented by the symbol \( k \), is a crucial value that tells us how stiff or flexible a spring is. Think of it as the spring's resistance to being stretched or compressed. The larger the spring constant, the harder it is to stretch the spring. This is defined by Hooke's Law, which says that the force \( F \) needed to extend or compress a spring is directly proportional to the distance \( x \) the spring is stretched or compressed: \( F = kx \).

To find \( k \), rearrange the formula to \( k = \frac{F}{x} \). In our exercise, the force provided was \( 13 \mathrm{~N} \) and the spring was stretched by \( 0.095 \mathrm{~m} \). By substituting these values into the formula, we calculate the spring constant \( k \) to be approximately \( 136.84 \mathrm{~N/m} \).

This tells us that for every meter this particular spring stretches, approximately \( 136.84 \mathrm{~N} \) of force is required. This stiffness measure helps us understand the spring's behavior in various applications.

Potential Energy

Potential energy is a type of energy that is stored within an object. Just waiting to do something! It is a measure of how much work the object can do due to its position or state. Different forms of potential energy include gravitational potential energy, elastic potential energy, and chemical potential energy.

In our exercise, we're interested in the potential energy stored in a spring as it is stretched. This energy can do work if the spring is allowed to return to its original state. The formula for the potential energy \( PE \) in a spring is given by:

• \( PE = \frac{1}{2} k x^2 \)

This formula captures how the energy stored increases with both the spring constant \( k \) and the square of the amount of stretch \( x \). As you stretch the spring more or use a stiffer spring, the potential energy increases because the spring can do more work when released.

Elastic Potential Energy

Elastic potential energy is a form of potential energy that is specifically stored when materials are stretched or compressed from their original shape, such as a stretched spring. The energy is "elastic" because it can be recovered when the object returns to its original form.

In our specific example, elastic potential energy is stored in the stretched spring. When the spring is released, it can convert this stored energy back into kinetic energy, possibly moving an attached object. The elastic potential energy \( E_{PE} \) in a spring can be calculated using the formula:

• \( E_{PE} = \frac{1}{2} k x^2 \)

Here, \( k \) is the spring constant, telling us how stiff the spring is, and \( x \) is how much the spring is stretched or compressed.

In our problem, substituting the given values, the elastic potential energy stored in the spring is \( 0.617 \mathrm{~J} \). This energy is ready to be unleashed as the spring returns to its normal length, returning to its state of rest.