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.
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.)
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