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Chapter 6: Problem 12

Determine whether the statement is true or false. Explain your answer. It follows from Hooke's law that in order to double the distance a spring is stretched beyond its natural length, four times as much work is required.

Short Answer

Expert verified
The statement is true; doubling the displacement requires four times the work.

Step by step solution

01

Understanding 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. Mathematically, this is represented as \( F = kx \), where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement from the natural length.
02

Work Done on a Spring

The work done on a spring is given by the integral of the force over the displacement. The formula is \( W = \int F \, dx \). Substituting Hooke's law, the equation becomes \( W = \int kx \, dx \), which evaluates to \( W = \frac{1}{2}kx^2 \). This describes the work required to stretch or compress a spring by a distance \( x \).
03

Comparing Work for Two Distinct Displacements

Let \( x_1 \) be the initial displacement that results in work \( W_1 = \frac{1}{2}kx_1^2 \). If the displacement is doubled, the new displacement is \( x_2 = 2x_1 \), resulting in work \( W_2 = \frac{1}{2}k(2x_1)^2 = \frac{1}{2}k(4x_1^2) = 2kx_1^2 \).
04

Analyzing the Work Comparison

Compare \( W_2 \) with \( W_1 \). We find \( W_2 = 4 \cdot \left(\frac{1}{2}kx_1^2\right) = 4W_1 \). Thus, doubling the displacement requires four times the initial work.
05

Conclusion

Since \( W_2 = 4W_1 \), the statement is true as per Hooke鈥檚 Law. Doubling the displacement indeed requires four times the work.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spring Constant
The spring constant, denoted as \( k \), is a fundamental concept in Hooke's Law. This constant represents how stiff or rigid a spring is. The higher the spring constant, the harder it is to stretch or compress the spring.

In mathematical terms, Hooke's Law is expressed as \( F = kx \), where \( F \) is the force applied to the spring, and \( x \) is the displacement from its natural length.
  • If \( k \) is large, more force is needed for the same displacement.
  • For a small \( k \), less force is required to stretch the spring.
Understanding the spring constant helps in predicting how much force is needed for a given stretch or compression.
When solving problems involving springs, always remember that \( k \) is key to linking force with displacement.
Work-Energy Principle
The work-energy principle forms a vital link between the effort required to stretch or compress a spring and the resulting work done.
The work done on a spring can be determined using an integral that accounts for the force over the displacement. To derive this work, we start with Hooke's Law which states \( F = kx \). Integrating this force from 0 to \( x \) gives us:\[ W = \int kx \, dx = \frac{1}{2}kx^2 \]This formula \( W = \frac{1}{2}kx^2 \) represents the work done when a spring is stretched or compressed from its natural length.
  • The work done depends on both \( k \) and \( x^2 \).
  • This quadratic relationship implies that doubling the displacement makes the work increase by four times, as shown when \( W \) changes with \( x^2 \).
Understanding this principle clarifies why more effort is required for larger displacements.
Displacement
Displacement in the context of springs refers to the distance a spring is stretched or compressed from its natural, or resting, length. It is represented by the variable \( x \) in Hooke's Law.
Displacement is key in calculating how the force applied to a spring translates into work.
  • A larger displacement \( x \) results in more force required, in accordance with \( F = kx \).
  • Doubling the displacement \( x \) doesn't just double the required work. Due to the \( x^2 \) term in the work formula, \( W = \frac{1}{2}kx^2 \), quadruple the work is necessary.
Understanding displacement allows for accurate predictions of the work needed for both small adjustments and major changes in a spring's length.
It highlights the non-linear increase in work compared to linear increases in displacement.

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