# Q & A: Conservation of energy in a system

Q:
My science group is doing a project on mechanical energy. I need some ideas on potential and kinetic energy at work. I also need information on the defintion of kinetic and potential energy. Please email me A.S.A.P. Thank you!
- Trent
Brier Terrace Middle School, W.A., U.S.A.
A:

Definitions first: Kinetic energy is the energy of things moving. The faster something moves, the more kinetic energy it has. (The frormula for kinetic energy is (1/2)mv^2, where m is the object’s mass and v is its velocity.) Potential energy is a bit trickier - you can think of it as energy that’s stored up, giving something the potential to do something. For example, if you lift something up, you give it what’s called ’gravitational potential energy’, or the potential to move when you let go of it. (The formula for gravitational potential energy near the Earth's surface is mgh, where m is the mass, g is the acceleration of gravity, and h is how high up the object is.)

The thing about energy is that it’s always conserved - that is, between potential and kinetic, you always have the same total amount of energy all together. So when you drop the book, what used to be gravitational potential energy turns into kinetic energy as it falls.

A pendulum would also work very well for this. You could start by pulling the pendulum up some distance. This will give it a certain amount of potential energy due to the force of gravity. Then let go of the pendulum. As the pendulum moves down, the gravitational potential energy is turned into kinetic energy. Almost all of the potential energy is turned into kinetic energy as the pendulum passes through the lowest point that it can go. (We say "almost" because there is always a little friction, so that a little bit of the energy goes to other foms, heating things up.) On its way back up, gravity does work on the pendulum to turn the kinetic energy to potential energy again. The pendulum therefore goes to almost the exact same height from which you let it go. This is because the potential energy that initailly was turned into kinetic energy is then turned back into potential energy without the energy being used anywhere else. This means that the total energy of the system is conserved. This shows the relation between kinetic and potential energy very well and can be used in most systems where there is some type of oscillation is taking place.

In cases where more friction is present, the sum of obvious kinetic and potential energy will not stay nearly constant. If you are driving in a car, for example, and put on the brakes, then the car will eventually stop. The kinetic energy in this case goes from a large value to zero, and the potential energy does not change as long as the road is horizontal.
It is still true that the total energy is conserved, however. The missing energy here just goes into heating things up. The brakes of a car get very hot when you stop. If we add up kinetic energy plus potential energy plus the heat energy, then we would again find that energy is conserved.

That heat is really just the kinetic energy of the atoms inside jiggling around and the potential energy of them squashing together and pulling apart. Technically, scientists call this thermal energy and mean something a little different by "heat", but often we forget and use the word this way.

I hope that this helps you guys come up with ideas.

(published on 10/22/2007)

## Follow-Up #1: pendulum and energy conservation

Q:
When I tried the pendulum experiment described in the response, my calculations (in Joules) for potential energy and kinetic energy we no where close to each other. I calculated kinetic energy using the velocity of the swing of the pendulum. What could I be doing wrong? Please help, my 8th graders are counting on this. Thank you.
- Jenn (age 38)
Mundelein, IL, USA
A:

Jenn- I'm very interested in getting to the bottom of this.   Thanks for sending the follow-up description of your experiment ()
and the data you got. For other readers, the pendulum weight was supposed to swing down about h=0.40 m and near the bottom of its swing traveled about 0.50 m in a time of about 1.55 s, giving v = 0.32 m/s. Then just from energy conservation, you expect the velocity at the bottom to be (2gh)1/2 which comes out about 2.8 m/s.

So the final speed seems low by almost a factor of ten, leaving the kinetic energy low by almost a factor of 80. You sent me nice data on the forward and backward swings, showing that the pendulum didn't slow down a whole lot, so friction isn't the problem.

My initial guess of what the problem may have been was that the initial height was only 0.4 cm rather than 0.4 m. That doesn't work out, however, because for a reasonable pendulum length (L about 2m?) it wouldn't give a big enough swing to cover the 0.5 m between the timing marks.

Looking at those times you sent, the total time to return to the first mark, on the side where the pendulum started, was about 3 s. That would be most of a complete swing, which indicates that the period of a complete swing was a bit longer than 3 s, which makes sense for a pendulum about 2m in length L.  (period= 2π(L/g)1/2) So here's what I suspect happened. Maybe the starting point was only 0.4 m sideways from the bottom of the swing, not 0.4 m up. If your pendulum is about 2.5 m long, I think the numbers would work out to be about what you observed. You want to just use the height above the bottom as h in your gravitational potential energy.

Let me know if this is right or wrong!

Mike W.

p.s. Jenn then wrote in that this was indeed the situation with the pendulum.

(published on 10/28/2014)