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Chapter 6: Problem 35

The forces required to extend a spring to various lengths are measured. The results are shown in the following table. Using the data in the table, plot a graph that helps you to answer the following two questions: (a) What is the spring constant? (b) What is the relaxed length of the spring? $$\begin{array}{llllll} \hline \text { Force (N) } & 1.00 & 2.00 & 3.00 & 4.00 & 5.00 \\\ \text { Spring length (cm) } & 14.5 & 18.0 & 21.5 & 25.0 & 28.5 \\\ \hline \end{array}$$

Short Answer

Expert verified
Short Answer: To determine the spring constant (k) and the relaxed length (\(x_0\)) of the spring, plot the given data on a graph, fit a straight line through the points, and calculate the slope (m) and y-intercept (b) of the line. The spring constant (k) is equal to the slope (m), and the relaxed length (\(x_0\)) can be found by setting the force (F) to 0 in the equation F = k(x - \(x_0\)) and solving for \(x_0\).

Step by step solution

01

Plot the data

Plot the given data on a graph with force on the x-axis and spring length on the y-axis.
02

Find the linear relationship

Once the data is plotted, fit a straight line through the points, representing the relationship between the force and spring length. Calculate the slope and the y-intercept of the line. This can be done using the formula: $$m=\frac{y_2-y_1}{x_2-x_1}$$ where m is the slope and x and y are the x and y coordinates of any two points on the line.
03

Calculate the spring constant

Use Hooke's Law, which states: $$F=k(x-x_0)$$ where F is the force applied to the spring, k is the spring constant, x is the spring length when the force is applied, and \(x_0\) is the relaxed length of the spring. The slope of the line (m) will give us the spring constant (k), since the value of m represents the change in force divided by the change in spring length.
04

Calculate the relaxed length

Use the y-intercept of the line (b) to determine the relaxed length (\(x_0\)) of the spring. This can be done by setting the force, F, to 0 in the linear equation, F = k(x - \(x_0\)), and solving for \(x_0\). The y-intercept represents the point at which the force is 0, so we can rewrite the equation as: $$0=k(x-x_0)$$ Solve for \(x_0\) to find the relaxed length of the spring.
05

Answer the questions

Using the calculated spring constant (k) and the relaxed length (\(x_0\)), answer the two questions: (a) What is the spring constant? and (b) What is the relaxed length of the spring?

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Most popular questions from this chapter

Two springs with spring constants \(k_{1}\) and \(k_{2}\) are connected in series. (a) What is the effective spring constant of the combination? (b) If a hanging object attached to the combination is displaced by \(4.0 \mathrm{cm}\) from the relaxed position, what is the potential energy stored in the spring for \(k_{1}=5.0 \mathrm{N} / \mathrm{cm}\) and $k_{2}=3.0 \mathrm{N} / \mathrm{cm} ?$ [See Problem \(83(\mathrm{a}) .]\)
Sean climbs a tower that is \(82.3 \mathrm{m}\) high to make a jump with a parachute. The mass of Sean plus the parachute is \(68.0 \mathrm{kg} .\) If \(U=0\) at ground level, what is the potential energy of Sean and the parachute at the top of the tower?
A record company executive is on his way to a TV interview and is carrying a promotional CD in his briefcase. The mass of the briefcase and its contents is \(5.00 \mathrm{kg}\) The executive realizes that he is going to be late. Starting from rest, he starts to run, reaching a speed of $2.50 \mathrm{m} / \mathrm{s} .$ What is the work done by the executive on the briefcase during this time? Ignore air resistance.
The power output of a cyclist moving at a constant speed of $6.0 \mathrm{m} / \mathrm{s}\( on a level road is \)120 \mathrm{W} .$ (a) What is the force exerted on the cyclist and the bicycle by the air? (b) By bending low over the handlebars, the cyclist reduces the air resistance to \(18 \mathrm{N} .\) If she maintains a power output of \(120 \mathrm{W},\) what will her speed be?
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