Phy 201
Your 'cq_1_00.1' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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The problem:
You don't have to actually do so, but it should be clear that if you wished to do so, you could take several observations of positions and clock times. The main point here is to think about how you would use that information if you did go to the trouble of collecting it. However, most students do not answer these questions in terms of position and clock time information. Some students do not pause the video as instructed. To be sure you are thinking in terms of positions and clock times, please take a minute to do the following, which should not take you more than a couple of minutes:
• Pick one of the videos, and write down the position and clock time of one of the objects, as best you can determine them, in each of three different frames. The three frames should all depict the same 'roll' down the ramp, i.e. the same video clip, at three different clock times. They should not include information from two or more different video clips.
• For each of the three readings, simply write down the clock time as it appears on the computer screen, and the position of the object along the meter stick. You can choose either object (i.e., either the pendulum or the roll of tape), but use the same object for all three measurements. Do not go to a lot of trouble to estimate the position with great accuracy. Just make the best estimates you can in a couple of minutes.
Which object did you choose and what were the three positions and the three clock times?
answer/question/discussion: ************************ I chose the pendulum from the third video. My three positions were 1 inch, 12 inches, and 15 inches. The time for 1 inch was 28.795 seconds, the time for 12 inches was 19.343.seconds, and the time for 15 inches was 29.562 seconds. It would be interesting if this could be recorded more accurately and if there were a way to do so. Also if done in person, the clock times may be a little more accurate. Often when I stopped the video the clock was between times.
In the following you don't have to actually do calculations with your actual data. Simply explain how you would use data of this nature if you had a series of several position vs. clock time observations:
• If you did use observations of positions and clock times from this video, how accurately do you think you could determine the positions, and how accurately do you think you would know the clock times? Give a reasonable numerical answer to this question (e.g., positions within 1 meter, within 2 centimeters, within 3 inches, etc; clock times within 3 seconds, or within .002 seconds, or within .4 seconds, etc.). You should include an explanations of the basis for your estimate: Why did you make the estimate you did?
answer/question/discussion: ************************ The clock was stated as being accurate to .01 seconds so there is the limit of this in determining the accuracy of my measurements. Determining measurements can be done fairly accurately because it is easy to stop the recording. It can be a little difficult, however, because the videos were somewhat blurry. I would say that we can accurately record the inches the object traveled to 0.5 inches. It was fairly easy to determine where the object was inch-wise, but it was more challenging to determine where the object fell within the inch. I would say that the times can be accurate within .05 seconds. While the clock is accurate to the .01 seconds, when I stopped the camera it often came up as being between recordings. Therefore I feel there should be a bit more error. I was wondering what the exact error of the clock was and also if there was a better way of determining the estimated error then my “logical” efforts.
Very good. Note that the blurriness is intentional. At some level everything we observe is in fact 'blurry'.
answer/question/discussion: ************************
• How can you use observations of position and clock time to determine whether the tape rolling along an incline is speeding up or slowing down?
answer/question/discussion: ************************ You could use the positions vs. time to calculate the speed at different locations on the ramp. For example, you could take the initial location of the roll of tape (A) and a position in the middle (B). You could then record the time it took to get to position B from position A. You could then find a location further down the ramp that was the same distance as from B as A and mark this position C. You could then record the time it took to get to position C from B. The shorter time would show that the tape was speeding up as if went down the incline and therefore had a greater speed. To calculate speed, you would use the equation speed=distance/time. 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?
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. 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.
• How can you use observations of position and clock time to determine whether the swinging pendulum is speeding up or slowing down?
answer/question/discussion: ************************ You could use the same technique as with the tape except it may be more helpful to use more points to get an accurate view of how a pendulum accelerates. For example, as the pendulum first moves down the ramp it should accelerate until it reaches its minimum point. We can call this moving from point A to point B. Point B should be somewhere in the middle of the ramp and point A should be the initial point. We can then record the average speed from point A to point B by taking the distance between the two and dividing it by time. It would be more helpful, however, to split the ramp into more segments. Point A and B could be split by point C. We can then see if the pendulum’s speed is increasing from point B to C from its speed from point A to C. The pendulum would then slow down after passing point B so it would be useful to record the speed at different points past B using the same equation. 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.
The strategies I outlined previously would show this behavior, within the limits of our accuracy in observation.
The pendulum does 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.
answer/question/discussion: ************************
• Challenge (University Physics students should attempt answer Challenge questions; Principles of Physics and General College Physics may do so but it is optional for these students): It is obvious that a pendulum swinging back and forth speeds up at times, and slows down at times. How could you determine, by measuring positions and clock times, at what location a swinging pendulum starts slowing down?
answer/question/discussion: ************************ You could make a graph of the velocities and then use this graph to determine the point in which speed begins to decrease.
answer/question/discussion: ************************
• Challenge (University Physics students should attempt answer Challenge questions; Principles of Physics and General College Physics may do so but it is optional for these students): How could you use your observations to determine whether the rate at which the tape is speeding up is constant, increasing or decreasing?
answer/question/discussion: ************************ 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
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).
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40 minutes
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I spent a lot of time figuring out the requirements for such a question, so the time may be a little higher then normal.
You've asked some great questions and expressed some fine insights, well beyond the scope of what is expected here but of course very welcome.
At some point in the near future we can discuss how you might bypass some of the more repetitive tasks in this course. These tasks are necessary for most students, but could get very tedious for someone with your apparent background and ability. Stick with it for a few assignments until we can fully assess what you do and do not need.
See my notes and let me know if you have questions.