Total mechanical energy - The total mechanical energy is the KE + PE. It will also go to 0 twice in each oscillation - at the endpoints where the mass is at its maximum extension or compression. 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. Since we've chosen graph E to be our velocity, our KE curve must look like the square of that. Potential energy - This takes a bit of analysis. 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). 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. This means that we can only consider the graphs D and E. Velocity - In contrast to the position graph, the velocity oscillates around 0. 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.Ģ. Note that we didn't know whether it was moving left or right or whether it was at its maximum compression or not. (Or N -"none", but C satisfies everything we know about the motion, so it works.) Graph A is constant - it's not oscillating at all - so that rules out A. This is true of graph C but not of graph B, so we can rule out B. (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. So all those graphs that go to zero or below must be ruled out. 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. How can we figure out where it should be at the start of the graph and what the graph should look like? That means it's been oscillating for a while when our clock starts so $t=0$ is NOT the maximum start. And it says this is when the sonic ranger starts. 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. Position - This problem is really about figuring out how the physical situation matches with the graph. 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.ġ. If none work, choose N.Īlthough 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. Note that the horizontal axis (with arrow) goes through the zero of the vertical axis in each case. 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. All the graphs are shown as functions of time (horizontal axis). 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. For the time interval shown, friction and other resistive forces can be ignored. At the time at which that happens, the cart looks as shown in the picture above. After a few oscillations the ranger starts to collect data. The cart has been pulled out to a positive x value greater than the rest length and released. 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 x-coordinate reported by the ranger is shown above the diagram. Presenting a sample problemĪ sonic ranger is attached to a spring, which is attached to a wheeled cart. Let's try working one such problem through in detail to see how it works. This becomes especially effective for more complex motions like oscillations. 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.
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