Problem 78 - Calculus with Concepts in Calculus

Question

According to Hooke's Law, the force \(f(x)\) exerted by a spring that has been expanded a distance \(x\) is given by \(f(x)=-k x\), where \(k\) is a positive constant called the spring constant. Find a function \(U\) such that \(f(x)=\) \(-d U / d x\). (Such a function \(U\) is called a potential energy function for the force \(f\). If \(U(0)=0\), then \(U(x)\) represents the amount of energy stored in the spring when it is expanded a distance \(x\).)

Step-by-Step Solution

Answer

The potential energy function is \( U(x) = \frac{k}{2}x^2 \).

Step 1: Understand Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the distance it is stretched, represented by the formula \( f(x) = -kx \). Here, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.

Step 2: Establish the Relationship between Force and Potential Energy

The problem requires finding a potential energy function \( U(x) \) such that the force \( f(x) = -\frac{dU}{dx} \). This means that the derivative of the potential energy function with respect to \( x \) gives us the negative of the force at \( x \).

Step 3: Integrate to Find U(x)

To find \( U(x) \), we need to integrate \( -f(x) \) with respect to \( x \). Since \( f(x) = -kx \), we have:\[U(x) = \int -(-kx) \, dx = \int kx \, dx.\]

Step 4: Perform the Integration

Compute the integral \( \int kx \, dx \) to find \( U(x) \):\[U(x) = \frac{k}{2}x^2 + C,\]where \( C \) is the constant of integration.

Step 5: Apply the Initial Condition \( U(0) = 0 \)

We use the initial condition \( U(0) = 0 \) to find the constant \( C \):\[U(0) = \frac{k}{2}(0)^2 + C = 0.\]This gives \( C = 0 \). Therefore, \( U(x) = \frac{k}{2}x^2 \).

Key Concepts

Potential Energy Function

The potential energy function, often denoted as \( U(x) \), is essential when dealing with Hooke's Law. It's a concept used to describe the energy stored within a spring when it is either compressed or extended.
This energy is called potential energy because it has the potential to perform work when the spring returns to its relaxed or equilibrium position. To find the potential energy function, we use the relationship that the force exerted by the spring, \( f(x) \), is the negative derivative of \( U(x) \). Mathematically, this is expressed as \( f(x) = -\frac{dU}{dx} \).
This relationship indicates that the energy stored in the spring, represented by \( U(x) \), changes as the spring force varies with distance \( x \).
When a spring is extended a distance \( x \), the potential energy stored in it at that moment can be represented as \( U(x) \). Providing \( U(0) = 0 \) specifies that when the spring is unstretched, no potential energy is stored.

Spring Constant

The spring constant, symbolized as \( k \), plays a crucial role in Hooke's Law, capturing the stiffness of a spring. This constant determines how much force is required for a certain amount of stretch or compression. A higher \( k \) value means a stiffer spring, which requires more force to change its shape.
In the equation \( f(x) = -kx \), \( k \) is a positive constant derived from the material properties of the spring and its construction. It quantifies the relationship between force applied and resultant displacement within the elastic limit of the spring.
Understanding \( k \) helps predict how the spring behaves under different conditions. If we double the spring constant, for a given displacement \( x \), the force exerted by the spring will double as well.
This concept not only helps calculate potential energy but also plays a role in designing systems where springs are a key component.

Integration of Force

Integration is a powerful tool used to find the potential energy function \( U(x) \) from the force function \( f(x) \). Given that \( f(x) = -kx \), to determine \( U(x) \), we perform integration with respect to \( x \) on the force equation's negative counterpart.
This process is represented mathematically as:

• First, set up the integral \( U(x) = \int -(-kx) \, dx = \int kx \, dx \).
• Compute the integral: \[ U(x) = \frac{k}{2}x^2 + C \] where \( C \) is the constant of integration.

Integration essentially sums up all small changes in force over the distance \( x \), providing us with the total potential energy stored.
To determine the constant \( C \), use the initial condition \( U(0) = 0 \):
Plugging this into the equation: \[ U(0) = \frac{k}{2}(0)^2 + C = 0 \], we find \( C = 0 \).
Thus, the final expression for the potential energy function is \( U(x) = \frac{k}{2}x^2 \), which provides insight into how energy changes as the spring is manipulated.