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Submit a copy of your original submission, inserting revisions and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).

• The instructions told you to pause the video multiple times.  Sometimes student miss this suggestion.

If you haven't used the 'pause' and 'play' buttons on your media player, you should go back and do so.

• The questions are phrased to ask not only what you see when you play the video, but what you see when you pause the video as instructed, and what you think you could determine if you were to actually take data from the video.  You aren't asked to actually take the data, but you need to answer how you would use it if you did.

It's fine if you have given more general descriptions, which are certainly relevant.  But answers to the questions should include an explanation of how you could use the series of position and clock time observations that are possible with this video.

• The questions also ask how much uncertainty there would be in the positions and clock times observable with this specific video.  Different people will have different answers, and some reasonable answers might vary from one clip to the next, or from one part of a clip to another.  However the answers should include a reasonable quantitative estimate. 

You should have estimated the number of seconds or fraction of a second to within which you think the time displayed on the computer screen might be accurate (e.g., is it accurate to within 10 seconds of the actual clock time, or to within 1 second, within .1 second, maybe even within .01 or .001 second).  You might not yet know enough about the TIMER to give an accurate answer, but give the best answer you can.

You should also indicate a reasonable estimate of the number of inches or fraction of an inch to within which you could, if asked, determine the position of each object.

The questions posed here anticipate much of what we will do in the quarter of the course, and are most appropriate to University Physics students and other interested students with a calculus background, or at least a strong background in precalculus. 

These topics are taken up in later assignments, and nobody is expected to know them at this point.  So don't worry about it if you don't understand; but if you do understand then you will have a nice head start.

STUDENT QUESTION:

It would be interesting to solve this question using acceleration instead of the crude changes in speed. I know there is a way to calculate this using changes in velocity over time, but how would this be done? How could this problem be solved using acceleration?

INSTRUCTOR RESPONSE:

You can't directly observe acceleration with this setup; you can only infer it from the motion. We will do so in upcoming experiments. There are two basic strategies:

1. Calculate velocities over short intervals and plot the velocity vs. the time at the midpoint of the interval. The result is a reasonable approximation of the v vs. t graph, especially when the acceleration is uniform; the slope of this graph is equal to the acceleration.

2. This one requires calculus so is mainly directed at University Physics students:  Do a curve fit for position vs. clock time. Take the second derivative of the equation of the resulting curve. This will be the acceleration function. This method depends on selecting the correct function for the curve. If acceleration is uniform then a second-degree (quadratic) polynomial is the appropriate choice; the problem with this is that the selecting a quadratic can be a self-fulfilling prophecy and you could miss any nonuniformity in acceleration.

ADDITIONAL STUDENT QUESTION

My question for this situation is where is the point that the pendulum begins to slow down? It is possible that the pendulum actually continues to increase in velocity down the entire ramp so it would be interesting to know where the exact point that the pendulum begins to slow.

INSTRUCTOR RESPONSE: 

The strategies I outlined previously would show this behavior, within the limits of our accuracy in observation.

The pendulum should indeed begin to slow after reaching its low point, as you conjecture. With accurate observations of position and clock time we could therefore confirm or reject this hypothesis.

STUDENT COMMENT (level of University Physics):

To determine whether the pendulum is speeding up at a constant, increasing or decreasing rate you could use a derivative. Derivatives describe the rate of increase or decrease of the rate so finding the derivative of the graph at a different points would allow you to determine whether the rate of the rate is constant or variable

INSTRUCTOR RESPONSE:

Excellent insight, and good idea, but our data might not give us a function to take derivatives of. The strategies I outlined previously will work for this situation. The appropriate curve fit for the motion observed here would be a sine or cosine function, which works to the extent that air resistance is negligible. (if we need a level of precision that requires us to take account of air resistance the analysis gets very much more complicated, and requires at least ordinary differential equations; we will see the differential equation for the undamped pendulum later in the course, but the damped case is beyond the scope of what we can do with just a calculus 1 and 2 prerequisite).

IMPORTANT CONCEPT:  Vertical distance between points and graph:

The figure below depicts three points and a graph.  This graph doesn't do a very good job of 'fitting' the three points, and it is unrelated to the Introductory Pendulum Experiment but can be used to clarify the meaning of 'the vertical distance between a point and the graph'.

The first of the three points lies below the graph, the second lies above the graph, the third lies below. 

The vertical distance between a point and the graph is how far that point lies above or below the graph.

Before reading further you should ask yourself which point lies furthest from the graph (i.e., furthest above or below the graph), and which lies closest.

The figure below depicts the first point and the segment of the graph above it.  There is also a short horizontal segment, which is the part of the x axis that lies above the point.  You should visualize the part of the above graph that is represented by this figure (in your mind, circle that part of the original graph).

The next figure depicts the second point and the segment of the graph below it.

The third figure depicts the third point and the segment of the graph above it.

It should be clear that the second point is much closer to the graph, in the vertical direction, than the first or the third.

It is more difficult to compare the first and the third.  The figure below depicts the three vertical distances, side by side, in decreasing order of distance between point and graph:

It seems clear that the first point is the furthest from the graph.

If our graph had a vertical scale we would be able to compare these vertical distances with the scale and estimate just how far each lies from the graph.

• We could then average the vertical distances, to get a measure of how well the graph fits the three points.  This average is called the average deviation or mean deviation.

• We could also find the mean and standard deviation of these distances, a reasonably simple process with which you might or might not not be familiar at this point of your education.  The standard deviation is  measure, similar to but not quite the same as the mean deviation, of how closely the graph fits the points, and it turns out that it is a better statistical measure than the average deviation.