Spring Potential Energy

55 Spring Potential Energy

We’ll do one more example of how to use conservation of energy, this time by looking at springs. As we did for gravity, first we will first figure out what the potential energy for a spring should be, and then use that to solve a problem.

Exercise 55.1: Spring Potential Energy

The force from a spring is given by $\vec F = -k\vec x$ where $\vec x=0$ is the equilibrium point.

A. Compute the work done by a spring force as a particle moves from $\vec x_{\rm i}$ to $\vec x_{\rm f}$ .

B. Plug in the work you computed into the work–energy theorem (i.e. $K_{\rm f}=K_{\rm i}+W$ ). Collect all f’s on the left hand side of the equation, and i’s in the right hand side of the equation. Use your result to determine what the definition of the potential energy of a spring should be.

We have just established that the potential energy of a spring is given by $U_{\rm spring}=\frac{1}{2}kx^2$ . This is another definition that you absolutely need to remember.

Let’s go ahead and use the conservation of energy equation for a spring to solve a simple problem.

Exercise 55.2: Spring Gun

A horizontal spring of constant $k$ is attached to one end of a tube in outer space (i.e. the only force in this problem is the spring force). A small mass $m$ is used to compress the spring a distance $d$ , and then the mass is let go. The mass is launched out of the tube.

Find the velocity at which the small mass is launched. If you get stuck, use the slider to reveal my diagram, and take it from there.

It will not surprise you that you can also combine energies. If you have a problem that involves gravity and a spring, then energy is conserved, so long as you add up all of the types of energies relevant to the problem. Let’s try it.

Exercise 55.3: Spring Launch

Consider now a vertical spring on Earth. The spring is in a tube, and the tube can fit a mass $m$ . We will defined the top of the spring at its equilibrium length as $y=0$ .

A. If we place the mass on the spring, after some bouncing, it will reach equilibrium. At that point, the spring is compressed some distance $d$ . Find $d$ .

B. Suppose you now push the mass an additional distance $D$ down, which cocks the spring. You use a trigger to let the spring go, which launches the mass $m$ up in the air. How high does the mass go? Remember $y=0$ is the top of the unstretched spring. If you get stuck, use the slider to reveal my picture and take it from there.

Key Takeaways

The potential energy of a spring is given by $\frac{1}{2}kx^2$ where $x$ is the distance that the spring is stretched or compressed.

55 Spring Potential Energy

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