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Chapter 10: Problem 1

A hand exerciser utilizes a coiled spring. A force of \(89.0 \mathrm{~N}\) is required to compress the spring by \(0.0191 \mathrm{~m}\). Determine the force needed to compress the spring by \(0.0508 \mathrm{~m}\).

Short Answer

Expert verified
The force needed is approximately \(236.34 \mathrm{~N}\).

Step by step solution

01

Understand Hooke's Law

The problem involves a spring, and Hooke's Law is fundamental to solving it. Hooke's Law states that the force required to compress or extend a spring by a distance \(x\) from its rest position is proportional to \(x\). It can be mathematically expressed as \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement of the spring from its rest position.
02

Calculate the Spring Constant (k)

From the given data, a force of \(89.0 \mathrm{~N}\) compresses the spring by \(0.0191 \mathrm{~m}\). We can use this information to calculate the spring constant \(k\). Rearrange Hooke's law to solve for \(k\): \( k = \frac{F}{x} \). Therefore, \( k = \frac{89.0}{0.0191} \approx 4654.5 \mathrm{~N/m} \).
03

Calculate the Force for the New Displacement

Using the spring constant \(k = 4654.5 \mathrm{~N/m}\) determined in Step 2, we now calculate the force \(F\) needed to compress the spring by \(0.0508 \mathrm{~m}\). Again, use Hooke's Law: \( F = kx = 4654.5 \times 0.0508 \approx 236.34 \mathrm{~N} \).
04

Finalize the Solution

The calculation from Step 3 gives us the force needed to compress the spring by \(0.0508 \mathrm{~m}\). This leads us to the final solution that a force of approximately \(236.34 \mathrm{~N}\) is required.

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Key Concepts

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

Understanding the Spring Constant
The spring constant, often represented by the symbol \(k\), is a unique value that defines how stiff a spring is. This constant is a vital component in applying Hooke's Law, as it relates to the force needed to displace the spring.
When you apply a force to a spring, it responds by either compressing or extending. The spring constant determines how much force you need for a unit of displacement.
In this exercise, we used the formula \( k = \frac{F}{x} \) to calculate \(k\). Here:
  • \(F\) is the force applied, which was initially \(89.0 \mathrm{~N}\).
  • \(x\) is the displacement of the spring, which was \(0.0191 \mathrm{~m}\).
Calculating \(k\), we found it was approximately \(4654.5 \mathrm{~N/m}\). This means that for every meter of displacement, the spring requires around \(4654.5 \mathrm{~N}\) of force.
Calculating the Force
Once you have the spring constant, calculating the force for any given displacement becomes straightforward using Hooke's Law, \( F = kx \).
Let's break it down:
  • \(k\) is the spring constant, already calculated as \(4654.5 \mathrm{~N/m}\).
  • \(x\) is the new displacement value, \(0.0508 \mathrm{~m}\).
By plugging these values into the formula, you get the force \( F = 4654.5 \cdot 0.0508 \approx 236.34 \mathrm{~N} \). This calculation shows that a greater force of approximately \(236.34 \mathrm{~N}\) is needed to achieve the new spring displacement.
Spring Displacement Insights
Spring displacement is the change in length when a force is applied. It can occur either by stretching or compressing the spring.
On a broader level, understanding displacement is crucial because it helps predict how a spring will react under various forces.
In our example:
  • Initial displacement was \(0.0191 \mathrm{~m}\), resulting in a specific force.
  • The new displacement was \(0.0508 \mathrm{~m}\), requiring a higher force.
As we see, increasing the displacement also results in a proportional increase in the force needed, following Hooke's Law. Understanding this relationship helps in designing and using spring-based systems, ensuring their operation within safe limits.

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Most popular questions from this chapter

An archer, about to shoot an arrow, is applying a force of \(+240 \mathrm{~N}\) to a drawn bowstring. The bow behaves like an ideal spring whose spring constant is \(480 \mathrm{~N} / \mathrm{m}\). What is the displacement of the bowstring?

A person who weighs \(670 \mathrm{~N}\) steps onto a spring scale in the bathroom, and the spring compresses by \(0.79 \mathrm{~cm}\). (a) What is the spring constant? (b) What is the weight of another person who compresses the spring by \(0.34 \mathrm{~cm}\) ?

A piece of mohair taken from an Angora goat has a radius of \(31 \times 10^{-6} \mathrm{~m}\). What is the least number of identical pieces of mohair that should be used to suspend a \(75-\mathrm{kg}\) person, so the strain \(\Delta L / L_{0}\) experienced by each piece is less than \(0.010 ?\) Assume that the tension is the same in all the pieces.

Atoms in a solid are not stationary, but vibrate about their equilibrium positions. Typically, the frequency of vibration is about \(f=2.0 \times 10^{12} \mathrm{~Hz},\) and the amplitude is about \(1.1 \times 10^{-11} \mathrm{~m}\). For a typical atom, what is its (a) maximum speed and (b) maximum acceleration?

A point on the surface of a solid sphere (radius \(=R)\) is attached directly to a pivot on the ceiling. The sphere swings back and forth as a physical pendulum with a small amplitude. What is the length of a simple pendulum that has the same period as this physical pendulum? Give your answer in terms of \(R\).
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