Problem 44 - University Physics with Modern Physics

Question

Half of a Spring. (a) Suppose you cut a massless ideal spring in half. If the full spring had a force constant \(k\) , what is the force constant of each half, in terms of \(k ?\) (Hint: Think of the original spring as two equal halves, each producing the same force as the entire spring. Do you see why the forces must be equal? (b) If you cut the spring into three equal segments instead, what is the force constant of each one, in terms of \(k ?\)

Step-by-Step Solution

Answer

Each half has a force constant of \( 2k \), each third has a constant of \( 3k \).

Step 1: Understanding Spring's Force Constant

The force constant, also known as spring constant, of a spring is a measure of its stiffness. For a spring with a constant \( k \), when it is stretched or compressed by a distance \( x \), it exerts a force \( F = kx \).

Step 1: Analyze the Full Spring

Imagine the original spring as composed of two equal halves when it is intact. The force produced by the full spring can be expressed as the sum of the forces exerted by each of these halves. So, for the whole spring, \( F = kx \).

Step 2: Analyze Cutting Spring in Two Halves

When the spring is cut into two equal halves, each half exerts the same force \( F = kx \) to maintain the same stiffness as the original spring. However, because these halves are shorter, each needs a greater force constant to exert the same force: \( F = 2kx_{1/2} \), thus the new force constant for each half becomes \( 2k \).

Step 3: Verify by Equating Forces

For each half-spring to produce the same stiffness as the whole spring, \( 2k \cdot \frac{x}{2} = kx \). This ensures that the half-spring needs a force constant of \( 2k \).

Step 4: Cutting Spring into Three Equal Segments

Similarly, if the spring is divided into three equal segments, the stiffness needed for each segment to maintain the total stiffness of the original is increased due to the reduced length. The force exerted by each needs to be \( 3k \cdot \frac{x}{3} = kx \). Hence the force constant for each segment is \( 3k \).

Step 5: Conclude with Results

For each half of the spring, the force constant is \( 2k \) and for each third of the spring, it is \( 3k \). This is consistent with the understanding that a shorter spring has a higher force constant.

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in physics that describes the behavior of springs. It defines the relationship between the force exerted by a spring and the amount it is stretched or compressed.

This relationship is expressed with the formula:

• \(F = kx\)

where:

• \(F\) is the force exerted by the spring,
• \(k\) is the spring force constant, and
• \(x\) is the displacement from the spring's natural, unstressed length.

Hooke's Law tells us that the force produced by a spring is directly proportional to its displacement.
This means that as you stretch or compress a spring more, the force it exerts increases. Understanding this concept is crucial, as it lays the foundation for analyzing any spring system, whether it's a toy, a vehicle suspension, or the spring in a pen.

Spring Stiffness

Spring stiffness, quantified by the spring force constant \(k\), is a measure of a spring's resistance to deformation. A spring with a high stiffness will require more force to stretch or compress than a spring with lower stiffness.

When a spring is cut into segments, its stiffness changes. For example, if you cut a spring into two halves:

• Each half will be shorter and thus stiffer.
• The force constant \(k\) of each half becomes \(2k\).

This happens because a shorter spring requires more force to achieve the same displacement as the original longer spring.
This concept reflects the calculation method described when cutting a spring into multiple equal parts, showing why the force constant increases with shorter segments.
Understanding spring stiffness is key for tasks such as designing mechanical systems, where precise control over force and movement is needed.

Ideal Spring

An ideal spring is a theoretical model that assumes the spring follows Hooke's Law perfectly without any deviations. This means:

• There is no energy loss due to heat or internal friction.
• The spring will not deform permanently, regardless of how much it's compressed or stretched, as long as it's within its elastic limit.

This idealization simplifies the study of springs and helps predict their behavior under different forces and displacements.

In real-world applications, all springs deviate slightly from the ideal spring model due to materials' imperfections and operating conditions.
However, understanding an ideal spring is vital because it provides a baseline from which real-world springs can be analyzed, allowing engineers and scientists to make necessary adjustments and predict behaviors in more complex systems.