Question

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

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

Plot the data

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

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.

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.

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.

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?