Question

A spring used in an introductory physics laboratory stores \(10.0 \mathrm{J}\) of elastic potential energy when it is compressed \(0.20 \mathrm{m} .\) Suppose the spring is cut in half. When one of the halves is compressed by \(0.20 \mathrm{m},\) how much potential energy is stored in it? [Hint: Does the half spring have the same \(k\) as the original uncut spring?]

Short answer

The potential energy stored in one half of the spring is 20.0 J.

Step-by-step solution

Understanding Spring Constant

The spring constant, denoted by \(k\), defines the stiffness of the spring. When the spring is cut in half, its new spring constant \(k'\) becomes double the original \(k\). So if \(k\) was the spring constant of the whole spring, \(k' = 2k\).

Elastic Potential Energy Formula

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 compression. For the original spring: \(U = 10.0\, \mathrm{J}\) and \(x = 0.2\, \mathrm{m}\).

Derive Original Spring Constant

Rearrange the formula to solve for \(k\): \[ k = \frac{2U}{x^2} \] Substitute \(U = 10.0\, \mathrm{J}\) and \(x = 0.2\, \mathrm{m}\): \[ k = \frac{2 \times 10.0}{0.2^2} = 500\, \mathrm{N/m} \]

Calculate Potential Energy for Half Spring

With \(k' = 2k = 1000\, \mathrm{N/m}\) for the half spring, calculate the potential energy when compressed by \(0.20\, \mathrm{m}\): \[ U' = \frac{1}{2} k' x^2 = \frac{1}{2} \times 1000 \times 0.2^2 \]

Perform Calculation for Half Spring

Substitute values into the energy formula: \[ U' = \frac{1}{2} \times 1000 \times 0.04 = 20.0\, \mathrm{J} \] Thus, the potential energy stored in one half is \(20.0\, \mathrm{J}\).

Key concepts

Spring Constant

The spring constant, often symbolized as \( k \), is a measure of a spring's stiffness. It quantifies how hard it is to compress or stretch the spring. When a spring is sturdy, it has a higher spring constant. The formula for elastic potential energy, \( U = \frac{1}{2} k x^2 \), shows us the role \( k \) plays in determining how much energy is stored when the spring is compressed by \( x \) meters. If you cut a spring in half, the stiffness of each piece increases, leading to a new spring constant of \( k' = 2k \). This is because the shorter length of the spring must sustain the same force over a smaller distance, effectively doubling its rigidity.

Compression

Compression describes the extent to which a spring is shortened from its natural length. When you compress a spring, you're doing work on it. The degree of compression is crucial because it helps dictate the amount of elastic potential energy stored. Using the formula \( U = \frac{1}{2} k x^2 \), we realize that the energy stored is proportional to the square of the compression distance \( x \). This means doubling the compression quadruples the stored energy, assuming the spring constant stays the same. In our solved problem, we compress the spring by 0.20 meters. Even when the spring is cut in half, this compression distance remains unchanged, allowing us to directly calculate changes in potential energy due to alterations in the spring's properties.

Elasticity

Elasticity refers to a material's ability to return to its original shape after being deformed by a force like compression or stretching. Springs are a common example of objects that display significant elasticity, allowing them to store energy and release it when they return to their original shape. This restoring force is what makes springs useful in mechanisms that need energy storage and release—such as in pens and watches. The elastic potential energy stored in a spring is reliant on its elasticity. A spring that is easier to compress has lower elasticity and a smaller spring constant, thereby storing less energy under the same compression. In our exercise, we see how slicing the spring enhances its apparent elasticity by doubling the spring constant, leading to more energy being stored despite using the same amount of compression.

Energy Transformation

Energy transformation is the process by which energy changes from one form to another. For instance, when you compress a spring, mechanical energy is transformed into elastic potential energy. This stored energy can later be released, converting back into kinetic energy as the spring returns to its original shape. In the context of our solved problem, compressing the spring stores potential energy. If released, this energy would transform back, propelling an object or the spring itself due to the restitution of its original form. The half spring, with its higher spring constant, transforms more mechanical energy into elastic potential energy upon compression, compared to the original. This demonstrates the profound impact altering the spring's physical structure has on energy transformation capabilities, highlighting the interconnected nature of these concepts.