Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

For any spring obeying Hooke's Law, show that the work done in stretching a spring a distance \(d\) is given by \(W=\frac{1}{2} k d^{2}\).

Step-by-Step Solution

Answer

Work done is \(W = \frac{1}{2}kd^2\).

Step 1: Understand Hooke's Law

Hooke's Law states that the force required to stretch or compress a spring by a distance \(x\) is proportional to \(x\). This is represented by the formula \(F = kx\), where \(F\) is the force, \(k\) is the spring constant, and \(x\) is the distance the spring is stretched or compressed.

Step 2: Determine Work Done

The work \(W\) done on the spring is the integral of the force applied over the distance the spring is stretched. The formula for work is \(W = \int F(x) \, dx\).

Step 3: Substitute Hooke's Law into the Work Formula

Using Hooke's Law, substitute \(F(x) = kx\) in the work formula: \(W = \int kx \, dx\) over the limits from \(0\) to \(d\), where \(d\) is the distance the spring is stretched.

Step 4: Integrate the Function

Integrate \(kx\) with respect to \(x\):\[W = \int_0^d kx \, dx = \left[ \frac{1}{2}kx^2 \right]_0^d \]. This results in \(W = \frac{1}{2}kd^2 - 0 = \frac{1}{2}kd^2\).

Step 5: Conclusion

The work done in stretching a spring by a distance \(d\) is \(\frac{1}{2}kd^2\), as shown by integrating Hooke's Law over the given distance.

Key Concepts

Work and Energy

In physics, the concepts of work and energy are fundamental in understanding mechanical systems. When we talk about work in the context of springs, we are referring to the amount of energy transferred by a force acting over a distance. This force, in the case of springs that obey Hooke's Law, is variable, which makes things a bit more interesting.

Work done on a spring is calculated as the integral of force over a distance. Specifically, this transfer of energy is done when stretching or compressing the spring. It's not enough to consider just the displacement or force alone. Again, both function together through the entire motion.

In terms of springs, when we achieve energy transfer by stretching it a distance, the system stores this energy as potential energy, and the formula for the work done becomes:

• Work \( W = \frac{1}{2} k d^2 \)

In this formula, \( k \) is the spring constant and \( d \) is the distance stretched or compressed. This relation shows how energy is stored in the spring, and retrieves this work as needed, behaving like a conservative force.

Integral Calculus

Integral calculus plays a pivotal role in calculating the work done when dealing with variable forces. In the case of springs, since the force isn't constant (it varies linearly with the displacement according to Hooke's Law), integration is necessary.

The key operation here is integrating the expression derived from Hooke's Law:

• Start with Hooke's Law: \( F(x) = kx \).
• Apply it to the work formula, \( W = \int F(x) \, dx \), setting the expression for \( F(x) \) into the integral.

This integral gives us the work done as:\[W = \int_0^d kx \, dx = \left[ \frac{1}{2}kx^2 \right]_0^d \]

Evaluating this definite integral provides the formula for work done, \( W = \frac{1}{2}kd^2 \). This process shows how integral calculus allows us to accumulate quantities over a range, essentially helping us understand the total work from the varying force applied during spring displacement.

Spring Constant

The spring constant, denoted as \( k \), is a fundamental property of a spring in Hooke's Law, highlighting the spring’s stiffness. It is the proportionality constant in the formula \( F = kx \), where \( F \) is the force required to stretch the spring, and \( x \) is the displacement from its original position.

• Springs with a higher value of \( k \) are stiffer and require more force to stretch them. Conversely, a lower \( k \) indicates a less stiff spring, easier to stretch.
• In the context of energy, \( k \) directly influences the amount of work done: \( W = \frac{1}{2}kd^2 \), meaning that for a given displacement \( d \), the work done increases with the stiffness of the spring.

Knowing the spring constant is crucial in engineering and physics, as it helps in designing systems that rely on predictable elastic properties, ensuring safety and functionality in practical applications like vehicle suspension systems or other mechanical devices that use or mitigate spring action.