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Chapter 6: Problem 23

You do \(8.5 \mathrm{J}\) of work to stretch a spring with \(k=190 \mathrm{N} / \mathrm{m},\) starting with the spring unstretched. How far does the spring stretch?

Short Answer

Expert verified
The spring stretches approximately 0.3 meters.

Step by step solution

01

Understand the variables

We know that the spring was unstretched at the beginning. The amount of work done, i.e., the energy used to stretch the spring is given as \(8.5 J (Joules)\). The spring constant is given as \(k = 190 N/m\). The issue is to find how far the spring gets stretched, i.e., the displacement (\(x\)).
02

Write down the knowns

Let's denote the displacement as \(x\). Then the work done, or energy stored in the spring is given by the formula for elastic potential energy which is \(E = \frac{1}{2} k x^2\). Substituting the known values into this formula, we get \(8.5 = \frac{1}{2} \cdot 190 \cdot x^2\).
03

Solve for \(x\)

If we solve the equation above for \(x\), we get \(x = \sqrt{\frac{8.5}{\frac{1}{2} \cdot 190}}\).

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Most popular questions from this chapter

A certain amount of work is required to stretch spring A a certain distance. Twice as much work is required to stretch spring B half that distance. Compare the spring constants of the two.

A force pointing in the \(x\) -direction is given by \(F=F_{0}\left(x / x_{0}\right)\) where \(F_{0}\) and \(x_{0}\) are constants and \(x\) is position. Find an expression for the work done by this force as it acts on an object moving from \(x=0\) to \(x=x_{0}\)

Spider silk is a remarkable elastic material. A particular strand has spring constant \(70 \mathrm{mN} / \mathrm{m},\) and it stretches \(9.6 \mathrm{cm}\) when a fly hits it. How much work did the fly's impact do on the silk strand?

A sprinter completes a 100 -m dash in 10.6 s, doing \(22.4 \mathrm{kJ}\) of work. What's her average power output?

A force given by \(F=b / \sqrt{x}\) acts in the \(x\) -direction, where \(b\) is a constant with the units \(\mathrm{N} \cdot \mathrm{m}^{1 / 2} .\) Show that even though the force becomes arbitrarily large as \(x\) approaches zero, the work done in moving from \(x_{1}\) to \(x_{2}\) remains finite even as \(x_{1}\) approaches zero. Find an expression for that work in the limit \(x_{1} \rightarrow 0\)
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