# University of Wisconsin Green Bay

A child is playing on a tall swing set at the park. The chain on her swing is about 4.8 m long. She wants her mother to push her fast enough that she can swing over the top of the swing set in a circle (with the chains taut.) How fast must her mother push her at the bottom of the swing in order for her to follow a circular path at the top?

• This is a two part problem. You need to look at both energy and forces in order to understand it.

It is perhaps most obvious to see the conservation of energy nature of this problem. As the swing goes around in a vertical circle, it changes speed. The swing is faster at the bottom of the motion, and gets slower as it moves to the top. As long as frictional forces are either known or small, any time you compare the speed of an object at two different locations, conservation of energy is the most direct way to understand the problem.

If you only recognized this as a conservation of energy problem, you would quickly discover that you need additional information. You can find the speed of the swing at the bottom of the circle by looking at energy only if you know the speed at the top of the circle. It cannot be zero, because if it was gravity would pull the swing straight down. So to learn the speed of the swing at the top of the circle, you are asked to relate motion (speed at the top of the circle) to force (gravity and tension). Force and motion of a single object are always related through Newton’s Second Law, so this is also a force or 2nd Law problem.

• ΣF=ma

The key equation for any problem that relates forces and motion is Newton’s Second Law. The first task in this problem is to find the speed of the swing at the top of the circle, which involves relating gravity and tension to centripetal acceleration.

KEb + PEb = KEt + PEt - Wnc

Once you know the speed of the swing at the top of the motion, you will relate it to the speed at the bottom of the motion through conservation of energy. The gravitational energy that is lost as the swing goes down in height is converted to kinetic energy. In this case, there is no work done by non-conservative forces, and so if your book does not include the Wnc term as it teaches conservation of energy, it is fine to leave it out at this stage.

Obviously, you can set up the problem in either order. However, you will have to solve ΣF=ma mathematically before you can fully solve KEb + PEb = KEt + PEt - Wnc.

• Step 1

Now that you know how fast the swing must move at the top of the circle in order to stay on a circular path while gravity pulls inward, you are ready to use conservation of energy to find the speed of the swing at the bottom of the circle.

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Step 2

The speed of the swing at the bottom of the circle is the information that was requested in this problem. No further mathematical solution is necessary.

• This problem asked you to think about the motion of the swing from both the perspectives of force and of energy.

Because the swing needs to move fast enough at the top of the circle to prevent falling straight down due to the effects of gravity, the first step is to look at force and motion and to find the minimum speed of the swing at the top of the arc. Words like minimum or maximum suggest that a passive force is at one of its extreme values. In this case, the slowest speed corresponds to the smallest possible tension of zero.

Because the gravitational energy that the swing loses as it moves downward in the circle becomes kinetic energy, we can use conservation of energy to find the speed of the swing at the bottom of the motion which would correspond to a speed of 6.9 m/s at the top.

In this case, the high speeds required for the swing (15 m/s is about 34 mph) explain why you couldn’t swing over the top of a swing set as a child no matter how hard you tried.