Problem 17 - University Physics with Modern Physics

Question

A spring stores potential energy \(U_0\) when it is compressed a distance \(x_0\) from its uncompressed length. (a) In terms of \(U_0\), how much energy does the spring store when it is compressed (i) twice as much and (ii) half as much? (b) In terms of \(x_0\), how much must the spring be compressed from its uncompressed length to store (i) twice as much energy and (ii) half as much energy?

Step-by-Step Solution

Answer

(a.i) 4U_0, (a.ii) U_0/4; (b.i) sqrt(2)x_0, (b.ii) x_0/sqrt(2)

Step 1: Understanding Potential Energy in Springs

The potential energy stored in a compressed spring is given by the formula \( U = \frac{1}{2}kx^2 \), where \( k \) is the spring constant, and \( x \) is the compression distance. In this exercise, \( U_0 = \frac{1}{2}kx_0^2 \) represents the potential energy when the spring is compressed by \( x_0 \). We will focus on using this formula to compare different energies and distances.

Step 2: Calculate Energy When Spring is Compressed Twice (Part a.i)

When the spring is compressed twice as much, the compression distance becomes \( 2x_0 \). Substituting into the potential energy formula, we have \( U = \frac{1}{2}k(2x_0)^2 = \frac{1}{2}k(4x_0^2) = 4 \left( \frac{1}{2}kx_0^2 \right) = 4U_0 \). Therefore, the potential energy stored is 4 times \( U_0 \).

Step 3: Calculate Energy When Spring is Compressed Half (Part a.ii)

For half the compression, the spring is compressed by \( \frac{x_0}{2} \). The potential energy calculation becomes \( U = \frac{1}{2}k \left( \frac{x_0}{2} \right)^2 = \frac{1}{2}k \left( \frac{x_0^2}{4} \right) = \frac{1}{4} \left( \frac{1}{2}kx_0^2 \right) = \frac{U_0}{4} \).So, the energy stored is \( \frac{U_0}{4} \).

Step 4: Determine Compression for Twice the Energy (Part b.i)

To find the compression that stores twice the energy, set \( U = 2U_0 \). Therefore, \( \frac{1}{2}kx^2 = 2 \left( \frac{1}{2}kx_0^2 \right) \). Solving for \( x \), \( x^2 = 2x_0^2 \) and \( x = \sqrt{2}x_0 \). Hence, the spring must be compressed by \( \sqrt{2}x_0 \) to store twice the energy.

Step 5: Determine Compression for Half the Energy (Part b.ii)

To find the compression for half the energy, set \( U = \frac{U_0}{2} \). Thus, \( \frac{1}{2}kx^2 = \frac{1}{2} \times \frac{1}{2}kx_0^2 \). This simplifies to \( x^2 = \frac{x_0^2}{2} \) and \( x = \frac{x_0}{\sqrt{2}} \). Therefore, the spring must be compressed by \( \frac{x_0}{\sqrt{2}} \) to store half the energy.

Key Concepts

Hooke's Law

Hooke's Law is a principle of physics that connects the force exerted by a spring to the distance it is stretched or compressed. It is mathematically expressed as \( F = kx \), where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement from its original, uncompressed length. According to Hooke's Law, the force required to compress or stretch a spring is directly proportional to the distance it is moved.

This law is relevant to our problem because it provides the basis for calculating potential energy stored in a spring. Remember, potential energy in a spring is due to the work done to compress or stretch it. When you know the spring constant \( k \) and the compression distance \( x \), you can predict how much energy is stored or how much force might be exerted by the spring.

• **Linearity**: The relationship between force and displacement is linear, as long as the material of the spring remains elastic.
• **Limitations**: Hooke's Law only applies until the spring is stretched to its limit of elasticity.

Spring Constant

The spring constant \( k \) is a measure of a spring's stiffness. It is a crucial part of Hooke's Law and is expressed in newtons per meter (\( ext{N/m} \)). The larger the spring constant, the stiffer the spring, and the more force required to compress or stretch it by a certain distance.

In our exercise, the spring constant plays a role in calculating the potential energy stored in the spring as it is compressed or stretched. The formula for potential energy \( U \) in a spring is \( U = \frac{1}{2}kx^2 \). Here, \( k \) determines how much energy is stored for a given displacement \( x \).

• **Determination**: The spring constant is determined experimentally by applying known forces to the spring and measuring the resulting displacements.
• **Impact on Energy**: Changes in the spring constant affect potential energy; a larger \( k \) results in more energy storage for the same displacement.

Energy Storage in Springs

Energy storage in springs relates to the potential energy that a spring has when it is compressed or stretched from its original shape. This potential energy is calculated using the formula \( U = \frac{1}{2}kx^2 \), where \( U \) is the stored energy, and \( x \) is the displacement from the spring's natural length.

In the given exercise, we explore how varying the compression distance changes the energy stored:

• **Twice the Compression**: When the spring is compressed twice as much, the potential energy becomes four times greater, \( 4U_0 \).
• **Half the Compression**: Compressing the spring half as much reduces the energy to one-fourth, \( \frac{U_0}{4} \).

To manipulate energy storage, the amount of compression or stretching can be adjusted, showcasing the power of potential energy calculation in predicting spring behavior. This concept highlights how energy can be controlled and utilized in many practical applications like vehicles' suspension systems or measuring devices.