Question

1. using the best - fit line, determine the spring constant of the spring. show all calculations, including the equation and substitution with units.

Explanation:

To determine the spring constant using the best - fit line, we use Hooke's Law, \( F = kx \), where \( F \) is the force applied to the spring, \( x \) is the displacement of the spring from its equilibrium position, and \( k \) is the spring constant. When we rearrange Hooke's Law to solve for \( k \), we get \( k=\frac{F}{x} \).

Step 1: Recall the data from the best - fit line

Let's assume that from the best - fit line of a graph (where the y - axis is force \( F \) and the x - axis is displacement \( x \)), we have two points or we can use the slope of the line. The slope of the line in a \( F - x \) graph is equal to the spring constant \( k \).

Suppose the equation of the best - fit line is \( y = mx + b \), in the context of \( F \) (y - variable) and \( x \) (x - variable), the equation is \( F=kx + b \). For an ideal spring, the y - intercept \( b = 0 \) (because when \( x = 0 \), the force \( F = 0 \)). So the equation simplifies to \( F = kx \), and \( k=\frac{F}{x} \).

Let's assume that from the best - fit line, when \( F = 10\space N \) (Newtons) and \( x = 0.05\space m \) (meters).

Step 2: Calculate the spring constant

We use the formula \( k=\frac{F}{x} \)

Substitute \( F = 10\space N \) and \( x=0.05\space m \) into the formula:

\( k=\frac{10\space N}{0.05\space m} \)

\( k = 200\space N/m \)

(Note: If you have actual data from the best - fit line, you should use that data. For example, if the best - fit line has a slope calculated from two points \((x_1,F_1)\) and \((x_2,F_2)\), the slope \( m=\frac{F_2 - F_1}{x_2 - x_1}\), and \( k=m\) since \( F = kx\) (with \( b = 0\) for the spring - force graph).)

Answer:

If we use the example data above, the spring constant \( k=\boldsymbol{200\space N/m}\) (the value will change depending on the actual data from the best - fit line).