Problem 26 - Calculus Early Transcendentals

Question

A spring has a restoring force given by \(F(x)=25 x .\) Let \(W(x)\) be the work required to stretch the spring from its equilibrium position \((x=0)\) to a variable distance \(x\) Graph the work function. Compare the work required to stretch the spring \(x\) units from equilibrium to the work required to compress the spring \(x\) units from equilibrium.

Step-by-Step Solution

Answer

Explain your answer using the work function and its graph.

Step 1: Understand the force function

The restoring force of the spring, given by Hooke's law, is \(F(x) = 25x\). The force function is linear, increasing with the stretch of the spring.

Step 2: Calculate the work function

It's important to note that the work required to stretch the spring is given by the integral of the force function, \(F(x)\), with respect to \(x\). So the work function, \(W(x)\), is given by: $$ W(x) = \int F(x) \, dx $$ Now, we should integrate the force function to find the work function.

Step 3: Integrate the force function

We have the force function, \(F(x) = 25x\). Integrate this function with respect to \(x\): $$ W(x) = \int F(x) \, dx = \int 25x \, dx $$ Doing the integration, we get: $$ W(x) = \frac{25}{2}x^2 + C $$ Since the spring is at its equilibrium position when \(x=0\), the work required to keep it at this position is zero. Therefore, \(C=0\).

Step 4: Write down the work function

Now we have the work function: $$ W(x) = \frac{25}{2}x^2 $$

Step 5: Graph the work function

The graph of the work function, \(W(x)=\frac{25}{2}x^2\), is a parabola. The parabola opens upward, with a minimum at \(x=0\). It is symmetrical about the y-axis.

Step 6: Compare the work required for stretching and compressing the spring

Since the graph of the work function \(W(x)\) is symmetrical about the y-axis, the work required to stretch the spring \(x\) units from equilibrium (\(W(x)\) when \(x>0\)) is equal to the work required to compress the spring \(x\) units from equilibrium (\(W(x)\) when \(x<0\)). In other words, compressing or stretching by the same amount of displacement takes the same amount of work.

Key Concepts

Work Function

The work function in the context of a spring is a crucial part of understanding mechanics. Work is the energy required to move an object against a force, in this case, the restoring force of the spring. The restoring force is given by Hooke's law, where the force function is expressed as a relationship between force and displacement. In the exercise, the force function is linear:

• Hooke’s Law: \[ F(x) = 25x \]
• The work function \( W(x) \) tells us the work done stretching or compressing the spring from equilibrium (\( x = 0 \)) to a distance \( x \).

The work function is derived by calculating the integral of the force function. The force function itself expresses how much force is needed based on displacement, while the work function gives the total energy expended in changing the spring's position.

Integration

Integration is a mathematical way of finding the total sum of a quantity, which in this case is the total work needed to stretch or compress a spring. It's like calculating the area under a curve of a graph represented by the force function.By integrating the force function \( F(x) = 25x \), we find the work function:

• \[ W(x) = \int 25x \, dx \]
• The result of the integration is:\[ W(x) = \frac{25}{2}x^2 + C \]

Here, \( C \) is a constant that represents the initial condition, and since no work is needed to keep the spring at equilibrium, \( C \) equals zero.This means our final work function, without the constant, is:

• \[ W(x) = \frac{25}{2}x^2 \]

Understanding integration helps to visualize the energy dynamics involved when dealing with physical systems such as springs.

Force Function

The force function describes how much force a spring generates or resists as it gets displaced from its equilibrium position. Based on Hooke's law, the force function is directly proportional to the displacement:

• \[ F(x) = 25x \]
• This function is linear, indicating a steady increase in force with increasing displacement.

The proportionality constant (25 in this case) represents the spring constant. It tells us how stiff or soft the spring is:

• A larger spring constant means a stiffer spring, which requires more force to achieve the same displacement.

Visualizing this force function helps us understand the physical properties and behavior of the spring system when it’s stretched or compressed.

Graphing Parabolas

Graphing the work function provides a visual tool to understand how the work varies with displacement. The work function is represented by a parabola:

• Equation: \[ W(x) = \frac{25}{2}x^2 \]
• The graph of \( W(x) \) is a curve that opens upward, showing that the work increases with more displacement.

This parabola is symmetrical around the y-axis, meaning:

• Stretching or compressing the spring by the same amount (\( x \) or \(-x \)) requires the same amount of work.

Understanding the shape and symmetry of the parabola helps compare energies required for compression versus extension in spring motion, making the concept of work intuitive and easily comprehensible.