# Labs from Chicago, Fall 1993 : Mechanical Potential Energy.

Dr. Rich Kron, Dr. Heidi Newberg, and Luisa Rebull
Labs written for the CARA Space Explorers, Fall 1993

This is meant to be handed out to the students.

## I. Introduction

Remember jumping on the bed when you were little? When you land on the bed, something pushes you back up. When you compress a spring, something inside the spring pushes back. And, if you stretch a spring, something pulls back. If you hold a small spring in your hand, it takes work to compress it -- some springs take a LOT of work to compress or stretch. But, as soon as you let the spring go, it bounces back to where it was before. So, you can think of "putting energy into the spring" when you compress or stretch it, and "getting energy back out" when you let it go. When you stretch the spring, you give it a form of potential energy, because it then has the potential to spring back to where it was. This kind of energy, the energy you put into the spring, is called mechanical potential energy.

So how can we get a number for the amount of energy that you put into a spring? How hard it is to compress or stretch is measured by a number called the spring constant. The spring constant is different for every spring and depends on what the spring is made out of, how tightly it's coiled, things like that. We symbolize the spring constant by a letter k. The units of k are g/s. If I compress a spring by an amount I'll call x, the energy contained in a spring is given by the formula:

MPE = (1/2) k x

Here x is the difference between the length the spring normally is, and the length it is when it is compressed. If I stretch, rather than compress, a spring by an amount x, the energy is still given by the amount above. Notice that the amount of energy contained in the spring is the same whether I stretch or compress the spring; if I compress the spring by 5 centimeters, it has the same energy as if I had stretched the spring by 5 centimeters. If I stretch the spring 5 cm, it wants to compress back to its original position; if I compress the spring 5 cm, it wants just as badly to stretch back to its original position.

We can convert mechanical potential energy to kinetic energy by compressing a spring with a mass on the end of it, then letting it go. If we point the mass towards the ceiling, then the kinetic energy will be converted to gravitational potential energy when it is as high as it will go. Then, the gravitational potential energy will be converted to kinetic energy as it falls to the floor. When it hits the floor, the energy is converted to heat and noise. Remember that the gravitational potential energy of something is given by:

GPE = m g h

where g = 980 cm/s, m is the mass of the object, and h is its height. Today we are going to work with the conversion of energy between gravitational potential energy and mechanical potential energy.

## II. Activities

Assemble your apparatus as we've shown you. Pick a mass to start with (you'll use other masses later). You need to measure several things before you put it all together:

What is the length of your spring before you compress it? (UNITS!)

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What is the length of your compressed spring? (UNITS!)

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So what is the difference between the relaxed and the compressed length? (UNITS!)

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What is the total mass of the things you are going to launch? (UNITS!)

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Go ahead and launch your system, trying not to hit anyone, and warning people you might hit before you let it go.

How high did the mass fly? (UNITS!)

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We know that the energy contained in the spring basically all goes into gravitational potential energy, so we can write this equation:

mgh = (1/2) k x

Rearrange this equation so that there is a "k = " on one side and a combination of letters on the other side. This is called "solving for k." Do it here:

Check with one of us to make sure you have it right, then stick in all the information you have to solve for the k of your spring. Remember g = 980 cm/s.

What is the k of your spring? (UNITS!)

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Now, pick a different mass. What is this new mass? (UNITS!)

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If you repeat the experiment with this new mass, how high will the mass rise? (UNITS!)

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Do it. Were you right (within experimental error)?

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If there is time, pick a new mass, and try again.

## III. Possible Quiz Questions

1. A spring has a spring constant k = 98,000 g/s2. When you hang 100 grams on the end, it stretches 1 cm. What is the energy contained in the spring? (UNITS!)

2. If a spring has 1,000,000 ergs contained in it when it is stretched 40 cm, what is the spring constant of the spring? (UNITS!)

3. Is there more energy stored in a spring when you hang a heavy weight on it or a light one?

4. How much energy is contained in a compressed spring right before it makes a 50 gram mass jump 50 cm? (UNITS!)

5. If a mass of 5 g is put on a spring launcher (with a k of 400,000 g/s2) that gets compressed 1 cm, how high will the mass jump?