In Figure 10-20, the magnitude of the force necessary to stretch a spring is plotted against the distance the spring is stretched.

Figure 10-20
(a) Calculate the slope of the graph, k, and show that F = kd, where k = 25 N/m.



(b) Find the amount of work done in stretching the spring from 0.00 m to 0.20 m by calculating the area under the graph from 0.00 m to 0.20 m.

(c) Show that the answer to part b can be calculated using the formula W = 1/2 kd2, where W is the work, k = 25 N/m (the slope of the graph), and d is the distance the spring is stretched (0.20 m). (Do this on paper. Your instructor may ask you to turn in this work.)

a) The slope of the graph is 25 N/m. To show that F = kd, we can use the equation F = m*a, where F is the force, m is the slope of the graph, and a is the acceleration. Since the acceleration is the change in velocity over time, and the velocity is the change in distance over time, we can substitute a = d/t, where d is the distance the spring is stretched and t is the time it takes to stretch the spring. Substituting this into the equation, we get F = m*(d/t). Since the time it takes to stretch the spring is constant, we can simplify this to F = kd, where k is the slope of the graph, which is 25 N/m.

b) The amount of work done in stretching the spring from 0.00 m to 0.20 m can be calculated by calculating the area under the graph from 0.00 m to 0.20 m. This area is equal to the area of a triangle with base 0.20 m and height 25 N/m, which is equal to 2.5 Nm.

c) To show that the answer to part b can be calculated using the formula W = 1/2 kd2, we can substitute the values for W, k, and d into the equation. W = 1/2 (25 N/m)(0.20 m)2 = 2.5 Nm, which is the same answer we got in part b.

(a) To calculate the slope of the graph, we can use the equation:

slope = ΔF / Δd

where ΔF is the change in force and Δd is the change in distance.

Given that k = 25 N/m, we can substitute this value into the equation:

slope = ΔF / Δd = k

Therefore, the slope of the graph, k, is equal to 25 N/m.

To show that F = kd, we can rearrange the equation:

F = ΔF + F_initial = kΔd + F_initial

Since we are considering the stretching of the spring, the initial force F_initial is 0. Therefore, we have:

F = kΔd

This shows that F = kd.

(b) To find the amount of work done in stretching the spring from 0.00 m to 0.20 m, we need to calculate the area under the graph from 0.00 m to 0.20 m.

The area under the graph represents the work done. In this case, the shape formed under the graph is a triangle.

The formula for the area of a triangle is:

Area = 1/2 * base * height

In this case, the base is the distance, which is 0.20 m, and the height is the force, which can be determined from the graph.

(c) To show that the answer to part (b) can be calculated using the formula W = 1/2 kd^2, where W is the work, k = 25 N/m (the slope of the graph), and d is the distance the spring is stretched (0.20 m), we can substitute the values:

W = 1/2 * k * d^2
= 1/2 * 25 N/m * (0.20 m)^2
= 1/2 * 25 N/m * 0.04 m^2
= 0.5 N * 0.04 m
= 0.02 Nm

Therefore, the work done in stretching the spring from 0.00 m to 0.20 m is 0.02 Nm.

(a) To calculate the slope of the graph, we need to determine the change in force (ΔF) and the change in distance (Δd) over a specific interval. In this case, the slope represents the spring constant (k), which is given as 25 N/m.

From the graph, choose two points on the line. Let's say the coordinates of these points are (d1, F1) and (d2, F2).

ΔF = F2 - F1
Δd = d2 - d1

Now, to calculate the slope (k), we use the formula:
k = ΔF / Δd

Substituting the values:
k = (F2 - F1) / (d2 - d1)

(b) To find the amount of work done in stretching the spring from 0.00 m to 0.20 m, we need to calculate the area under the graph from 0.00 m to 0.20 m.

The area under the graph represents the work done. Since the graph is a straight line, the area can be calculated as the area of a rectangle.

The formula for the area of a rectangle is:
Area = base * height

In this case, the base is the distance (d) and the height is the force (F).

Given: d1 = 0.00 m, d2 = 0.20 m, F1 = 0 N (at d = 0.00 m), F2 = k * d2

Using the formula for the area:
Area = (d2 - d1) * (F2 - F1)

Substituting the given values:
Area = (0.20 m - 0.00 m) * (k * d2 - 0 N)

(c) To show that the answer to part (b) can be calculated using the formula W = 1/2 kd^2, where W is the work, k = 25 N/m (the slope of the graph), and d is the distance the spring is stretched (0.20 m), we need to derive this formula.

The work done in stretching a spring can be calculated using the formula:
W = 1/2 k Δd^2

We can re-arrange this formula as:
W = 1/2 k (d2 - d1)^2

Substituting the given values:
W = 1/2 k (0.20 m - 0.00 m)^2
W = 1/2 k (0.20 m)^2
W = 1/2 k (0.04 m^2)
W = 1/2 (25 N/m) (0.04 m^2)
W = 0.5 N/m (0.04 m^2)
W = 0.02 N⋅m or J

Therefore, W = 0.02 Joules.