spring has k=82 N/m use a graph to determine the work needed to stretch it from x= 3.8 cm to x=6.2 cm where x is the displacement

To determine the work needed to stretch a spring from x=3.8 cm to x=6.2 cm, we can use the graph of the force-displacement relationship of the spring. Here's a step-by-step explanation of how to do it:

1. Start by understanding the force-displacement relationship of a spring. According to Hooke's Law, the force exerted by a spring is proportional to the displacement from its equilibrium position. Mathematically, this can be represented as F = -kx, where F is the force, k is the spring constant, and x is the displacement.

2. Plot the graph: On a graph, label the x-axis as "Displacement" and the y-axis as "Force." Since we are given that k = 82 N/m, it can be considered a known constant in this case.

3. Plot the data points: For this particular case, we want to plot points representing the displacements x=3.8 cm and x=6.2 cm on the x-axis.

4. Calculate the corresponding forces: Use Hooke's Law to calculate the forces at the two displacements. Recall that F = -kx, where F is the force, k is the spring constant, and x is the displacement. Plug in the known values of k and the displacements (x=3.8 cm and x=6.2 cm) to calculate the corresponding forces.

5. Connect the data points: Draw a straight line connecting the two data points on the graph. This line represents the relationship between the displacement and force for the given spring.

6. Calculate the work: To calculate the work done, we need to find the area under the line connecting the two data points. Since the force is negative, the work done will be the negative of the area under the line. The work done is given by the formula: work = force * displacement. Calculate the work done by multiplying the average force by the displacement between the two data points.

By following these steps and using the given information, you should be able to determine the work needed to stretch the spring from x=3.8 cm to x=6.2 cm using the graph of the force-displacement relationship.

To determine the work needed to stretch the spring from x = 3.8 cm to x = 6.2 cm, we can use the formula for work done on a spring:

W = (1/2) * k * (x2^2 - x1^2)

Given that k = 82 N/m, and the displacements are x1 = 3.8 cm and x2 = 6.2 cm, we can substitute these values into the formula:

W = (1/2) * 82 * ((6.2^2) - (3.8^2))

W = (1/2) * 82 * (38.44 - 14.44)

W = (1/2) * 82 * 24

W = 82 * 12

W = 984 N·cm

Therefore, the work needed to stretch the spring from x = 3.8 cm to x = 6.2 cm is 984 N·cm.