Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

A force of 6 pounds is required to keep a spring stretched \(\frac{1}{2}\) foot beyond its normal length. Find the value of the spring constant and the work done in stretching the spring \(\frac{1}{2}\) foot beyond its natural length.

Step-by-Step Solution

Answer

Spring constant is 12 pounds/foot, work done is \( \frac{3}{2} \) foot-pounds.

Step 1: Understand Hooke's Law

Hooke's Law states that the force required to stretch or compress a spring is directly proportional to the displacement from its natural length. It is represented by the formula:\[ F = kx \]where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement from the natural length. In our case, \( F = 6 \) pounds and \( x = \frac{1}{2} \) foot.

Step 2: Calculate the Spring Constant

Using Hooke's Law, plug the given values into the formula:\[ 6 = k \times \frac{1}{2} \]To solve for \( k \), rearrange the equation:\[ k = \frac{6}{\frac{1}{2}} = 12 \]So, the spring constant \( k \) is 12 pounds per foot.

Step 3: Understand Work Done on a Spring

The work done on a spring is calculated using the formula:\[ W = \int_{0}^{x} F \, dx = \int_{0}^{x} kx \, dx \]We will use this formula to find the work done in stretching the spring \( \frac{1}{2} \) foot beyond its natural length.

Step 4: Calculate the Work Done

Substitute the previously found spring constant \( k \) and the displacement \( x = \frac{1}{2} \) into the work formula:\[ W = \int_{0}^{\frac{1}{2}} 12x \, dx \]First, compute the integral:\[ W = 12 \left[ \frac{x^2}{2} \right]_{0}^{\frac{1}{2}} \]This becomes:\[ W = 12 \left( \frac{(\frac{1}{2})^2}{2} - 0 \right) = 12 \left( \frac{1}{8} \right) = \frac{12}{8} = \frac{3}{2} \]Thus, the work done is \( \frac{3}{2} \) foot-pounds.

Key Concepts

The Spring Constant

When dealing with springs, the spring constant plays a crucial role in understanding how "stiff" a spring is. The spring constant, denoted by the symbol \( k \), quantifies the relationship between the force applied to a spring and the resulting displacement it causes.
In Hooke's Law, this relationship is written as \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement from its natural position.

Imagine a spring attached to a wall. If you pull the spring and it stretches, the spring constantly pushes back harder as you stretch it more. The spring constant \( k \) tells us how much force is needed for each unit of distance you stretch the spring.

• A larger spring constant indicates a stiffer spring, requiring more force to stretch or compress it.
• Conversely, a smaller spring constant means the spring is more flexible and easily deformable.

For example, in our problem, we found that \( k = 12 \) pounds per foot. This means that it requires 12 pounds of force to stretch this spring by one foot, underlining the stiffness of this particular spring.

Work Done on a Spring

When you stretch or compress a spring, you're putting in energy—this energy is what we call "work." The work done on a spring is calculated through integration because the force you need to keep stretching the spring varies as the spring stretches more.

The formula for work done on a spring based on Hooke’s Law is given by:\[W = \int_{0}^{x} kx \, dx\]In this formula, \( x \) is the displacement from the natural length, and \( k \) is the spring constant.

In our exercise, we wanted to find out how much work is done when the spring is stretched by \( \frac{1}{2} \) foot. Using the given spring constant \( k = 12 \), we calculated the work as \( \frac{3}{2} \) foot-pounds.

• This means that if you apply \( 1.5 \) foot-pounds of effort, the spring will stretch \( \frac{1}{2} \) foot beyond its natural length.
• When calculating work, we're calculating the area under the force-displacement graph from 0 length to \( x \).

It's another way of seeing the direct relationship between the amount of force applied and the distance stretched.

Calculus Integration in Spring Problems

Calculus comes into play in spring problems when we need a precise calculation of work done over a period of stretching. Integration helps us to find this cumulative effect due to variable forces applied as you stretch the spring further.

In this context, the integral \( \int_{0}^{x} kx \, dx \) is computed to get the work done. Here’s a step-by-step breakdown:

• Plug in \( k \) and \( x \): You start with the integral structure and plug the specific values of the spring constant and the maximum displacement.
• Compute the integral: This step involves mathematical manipulation to solve the integral, which leads you to the result.
• Evaluate the result from 0 to \( x \): You're essentially calculating the "area under the curve," which in physical terms equates to the total work done by the force over that displacement.

In our example, using the values \( k = 12 \) and \( x = \frac{1}{2} \), the integral calculation gives us \( \frac{3}{2} \) foot-pounds of work. This computed figure represents the total effort needed over the \( \frac{1}{2} \) foot stretch. The principle of integration allows us to sum up infinitely small amounts of forces and their contributions, an essential arithmetic skill in physics and engineering.