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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
True, four times more work is needed to double the stretch.

Step by step solution

01

Understand Hooke's Law

Hooke's Law states that the force needed to stretch or compress a spring by some distance \( x \) from its natural length is directly proportional to \( x \). Mathematically, it is expressed as \( F(x) = kx \), where \( k \) is the spring constant.
02

Work Done on a Spring

The work \( W \) done by stretching or compressing a spring from an initial distance \( x_1 \) to a final distance \( x_2 \) is given by the integral \( W = \int_{x_1}^{x_2} kx \, dx \). This evaluates to \( W = \frac{1}{2} kx_2^2 - \frac{1}{2} kx_1^2 \).
03

Calculate Initial Work to Stretch Spring by \( x \)

If we stretch the spring from its natural length (\( x = 0 \)) to a distance \( x \), the work done is \( W_1 = \frac{1}{2} kx^2 \).
04

Calculate Work to Stretch Spring by \( 2x \)

To stretch the spring from its natural length to a distance \( 2x \), the work done is \( W_2 = \frac{1}{2} k(2x)^2 = 2kx^2 \).
05

Compare Work at Distances \( x \) and \( 2x \)

We compare \( W_1 = \frac{1}{2} kx^2 \) and \( W_2 = 2kx^2 \). Thus, \( W_2 = 4W_1 \), confirming that four times the work is needed to double the spring's stretch from \( x \) to \( 2x \).

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Key Concepts

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

Work Done by a Spring
When you stretch or compress a spring, you are performing work on it. This concept comes to life through Hooke's Law. To calculate the work done in stretching a spring, you use integration.
When a spring is stretched from an initial distance of zero to a final distance of some value, say \(x\), the work done, or energy required to do this, is captured by the equation \( W = \frac{1}{2} k x^2 \). The 'work' here essentially refers to the energy transferred to the spring as it is deformed.
  • The formula for work accounts for how far the spring is stretched and the spring's stiffness, expressed by the constant \( k \).
  • Calculating work involves integrating the force over the distance, which for a spring results in a quadratic formula.
For example, doubling the amount of stretch does not require double the work because the work grows quadratically with distance. Instead, it requires four times the work, as stretching from \( x \) to \( 2x \) changes the work from \( \frac{1}{2} kx^2 \) to \( 2kx^2 \).
Spring Constant
The spring constant, denoted by \( k \), acts as a measure of a spring's stiffness. It tells you how much force is needed to stretch or compress the spring by a certain distance.
A higher spring constant means a stiffer spring that requires more force to deform. This spring constant is unique to each spring and is a crucial part of understanding how that spring behaves under different loads.
  • The unit of the spring constant is Newton per meter (N/m), highlighting its role in force and displacement calculations.
  • The value of \( k \) helps predict how much energy or work is involved when manipulating the spring.
Understanding \( k \) allows you to apply Hooke's Law effectively, as it directly impacts the force required to achieve different displacements.
Force and Displacement Relation
According to Hooke's Law, the force exerted by a spring is proportional to its displacement from its natural, or un-stretched, position. This linear relationship is expressed as \( F = kx \), where:
  • \( F \) is the force exerted by the spring (in Newtons).
  • \( x \) is the displacement from the natural length (in meters).
  • \( k \) is the spring constant (Newton per meter).
This means that doubling the displacement doubles the force, assuming the spring constant remains the same.
However, keep in mind that this formula is a simplification valid for small displacements within the limits of elasticity. Once you surpass the elastic limit, the material may not obey this linear relationship. Therefore, Hooke's Law is a helpful way to understand the interaction between force and displacement for ideal springs.

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Most popular questions from this chapter

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