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Illustration 5.4: Springs and Hooke's law

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Springs are interesting objects that for a range of stretching and compression follow Hooke's law. Hooke's law states that the force that the spring exerts is F = -k Δx, where k is the spring constant and Δx is the displacement of the spring from its equilibrium position. In this Illustration the spring can be stretched by click-dragging the blue ball (position is given in centimeters and force is given in newtons). Slowly drag the spring back and forth and watch the graph. Restart.

Where is the equilibrium position of the spring? Given that Δx is measured from the equilibrium of the spring, look for the position where F = 0 N. This occurs at x = 30 cm. This is the equilibrium position.

What is the spring constant of the spring? It is not the force shown in the table divided by the position shown in the table. Why not? Recall that the "Δx" in the spring force equation is measured from equilibrium. Therefore, at maximum extension, x = 20 cm and the force is -160 N. Therefore, k = 800 N/m. Given that the negative of the slope of the line on the graph should also be k, we can measure the spring constant by finding the slope and we get the same result.

The fact that spring forces are variable with position means that while we can determine the force, we cannot (given what we currently know) determine the velocity and position vs. time for an object attached to a stretched spring. Why? The force is not constant (it varies with position), and therefore the acceleration is not constant. This means we cannot use the kinematic equations for constant acceleration. What can we do? We can use concepts that you will learn about in Chapters 6 and 7.

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