# Example: Oscillator graphs

## Understanding the situation

We've spent a lot of effort learning to translate a physical motion into a graph, look for consistency among those graphs, and interpret what information each graph gives us about the motion. This becomes especially effective for more complex motions like oscillations. Let's try working one such problem through in detail to see how it works.

## Presenting a sample problem

A sonic ranger is attached to a spring, which is attached to a wheeled cart. The x-coordinate reported by the ranger is shown above the diagram. The small black circle on the axis indicates the point at which the cart’s arrow points when the spring is at its rest length.

The cart has been pulled out to a positive x value greater than the rest length and released. After a few oscillations the ranger starts to collect data. At the time at which that happens, the cart looks as shown in the picture above. We take that time as $t = 0$. For the time interval shown, friction and other resistive forces can be ignored.

In the figures below are shown graphs for a number of different variables either taken directly from the ranger’s data or calculated from it. All the graphs are shown as functions of time (horizontal axis). Identify which of the graphs could possibly be graphs of the indicated variables for the same motion of the cart if the vertical axis had the right units. Note that the horizontal axis (with arrow) goes through the zero of the vertical axis in each case. Put all the answers that could be right. If none work, choose N.

1. x position
2. x velocity
3. potential energy
4. kinetic energy
5. total mechanical energy

## Solving this problem

Although the times are given on the graphs, there are no numbers for the spring constant or mass, so we can't calculate the period of oscillation. But it does say "for the same motion" so once we have chosen a motion by picking one graph, we have to demand that the others be consistent with it.

1. Position — This problem is really about figuring out how the physical situation matches with the graph. Notice that it says that the cart is started by pulling it out to "a positive x value greater than its rest length" but the picture shows the cart at a position at a positive x value LESS THAN its rest length. And it says this is when the sonic ranger starts. That means it's been oscillating for a while when our clock starts so $t=0$ is NOT the maximum start. How can we figure out where it should be at the start of the graph and what the graph should look like?

We know it will never oscillate all the way to zero for two reasons: first, the sonic ranger can't read down to zero, and second, you can't compress a spring to zero length. So all those graphs that go to zero or below must be ruled out. (Graphs D, E, F, G, H, and I) We also know that it will oscillate around the equilibrium point, equal amounts above and below, so since it is to the left of the equilibrium point it's below the midpoint of the oscillation. This is true of graph C but not of graph B, so we can rule out B. Graph A is constant — it's not oscillating at all — so that rules out A. The answer must be C. (Or N —"none", but C satisfies everything we know about the motion, so it works.)

Note that we didn't know whether it was moving left or right or whether it was at its maximum compression or not. But graph C tells us that it is not at its maximum compression at the instant the photo is taken and that it is moving to the left.

2. Velocity — In contrast to the position graph, the velocity oscillates around 0. This means that we can only consider the graphs D and E. Graph D says the velocity at time 0 is positive and then goes to 0, while graph E says the velocity at time 0 is negative and then goes to 0. Our position graph told us that the initial velocity was to the left, so negative, on its way to its maximum compression (and therefore a zero velocity). That means the answer must be E.

3. Potential energy — This takes a bit of analysis. The PE is given by PE = $\frac{1}{2}k(ΔL)^2$. But we have to interpret what $ΔL$ means in this physical system. It represent the stretch or squeeze of the spring, so it's the distance from the black dot. The PE is going to always be positive (it's a square of a number that could be positive or negative) so that rules out D and E. Every time the oscillation goes through the equilibrium point $ΔL$ goes through 0, so the graph must go down to zero. This suggests graphs F, G, H, and I But $ΔL$ goes through 0 at every midpoint of the oscillation — twice in each oscillation! (Once going each direction.) G and H only do it once so they are ruled out. That leaves F and I. Our position graph starts at a large displacement (compression) from the origin and keeps going — getting larger. So the PE at the start should be positive and growing. This looks like graph F. But of course our answer could be N (none)! So we have to check that F goes through 0 at every midpoint of the oscillation in graph C (it does) and is a maximum at every extreme point of graph C, both high and low (it is). So the answer must be F.

4. Kinetic energy — The analysis here is similar. The KE is given by KE = $\frac{1}{2}mv^2$. Since we've chosen graph E to be our velocity, our KE curve must look like the square of that. It will always be positive, start out small positive and it's slowing down on its way to its maximum, where the velocity is 0. It will also go to 0 twice in each oscillation — at the endpoints where the mass is at its maximum extension or compression. This matches graph I.

5. Total mechanical energy — The total mechanical energy is the KE + PE. Since we are told that resistive forces (friction, viscosity, air drag) can be ignored, total mechanical energy will be a constant: graph A.

Joe Redish 4/7/16

Article 667