Class 5: Horizontal Motion

QUIZ 2: readings CH 3, pages 34-43

How do we begin to move horizontally? Starting from a forward lunge or 4th position we push off the floor after raising our front leg forcing our center of gravity to no longer be over the area of support. It is important to recognize that the only force that can cause a solo dancer to move horizontally is friction from the floor. This is a reaction force to the dancer pushing against the floor in the opposite direction. Recall that we must consider all of the forces acting ON the object (as usual the dancer). This week we will use video to measure our position as we begin horizontal runs. From that position data we will calculate new quantities: velocity and acceleration. It is acceleration that results from net external forces as related in Newton’s 2nd Law, the sum of the forces equals mass times acceleration or ΣF=ma .

The first quantity that we define is displacement. We will often use a symbol (x, y, or H) to represent it. It is similar to a distance traveled with an important extra bit of information: direction. If we were to say that we leave San Francisco and drive 400 miles, that would be a distance. However a displacement of 400 miles to the southeast might put us in Los Angeles (or the ocean if we chose direction unwisely). We can calculate displacement by finding the difference in position between the final position (x_final) and the initial position (x_initial). To distinguish displacement from distance, imagine driving 400 miles around and around campus. If you start and stop from the same parking space your displacement will be zero even thought you traveled 400 miles.

“Velocity” is like speed except it also has the direction as well. It can be computed from the displacement per time or Δx/Δt . We use the symbol Δ to represent a change or difference in a quantity. Δx= x_final-x_initial is just a displacement. Δx is the change in time over which our is measured. For example our video cameras have a rate of 30 frames per second. The Δt for video data will then be 1/30 second or 0.03333 seconds. The maximum data rate for the force plates or 50 readings per second, so the corresponding Δt will be 1/50 or 0.02 seconds.

We will analyze video data of horizontal runs to produce displacement versus time and velocity versus time curves. From the velocity versus time, we can estimate (and occasionally calculate) the acceleration. Acceleration is a quantity which tells us how fast and in which direction the velocity is changing. It is defined as a=Δv/Δt , the change in velocity over the change in time. Acceleration is important because it is the directly connected to the force, the other measurable quantity in our experiments. This connection is apparent in Newton’s 2nd Law, ΣF=ma .

If we think about starting from 1st position, 4th position, or a sprinter’s stance; what are the expected differences in acceleration? Why? You might imagine that a greater horizontal force (friction) from the runner’s stance will lead to a greater acceleration. What will happen to the acceleration after you run for a long time? Does your velocity continue to increase? Probably not. We might expect that after some time the acceleration becomes about zero as the maximum velocity is achieved. It is important to note that we probably will not reach this state in the confines of a small dance studio. You might want to include a videotape of a run outdoors to observe acceleration going towards zero during a run.

Class 6:

We use a simple video of a run with our calibrated backdrop (or simply a meter stick on the wall) to generate a plot of our position vs. time. Using video software we view the run frame by frame. The first step is to determine how a distance (1 meter) in the dance studio compares to a distance on the computer screen used for viewing. Note that this relationship will depend on your monitor, the size of the video software frame on your computer display, and the particulars of your camera setup and zoom in the dance studio. For convenience, use the metric (centimeter with millimeter divisions) side of the ruler. Measure the length of a meter in the room. If it is say 7.5 cm, then your screen measurements can be converted 7.5 cm = 1 m. In other words, divide your screen measurements by 7.5 to get studio distances in meters. It is best to write down a list of your cm scale distances first in a column in LoggerPro, and then you can convert all at once (less likelihood of an error). If you videotaped running from right to left you might notice that your ruler distances get smaller. One can simply subtract those data from the largest position value to get increasing distance. LoggerPro allows you to now pick what variables you plot. Let’s choose position vs. time as shown in the upper frame figure 5.

Using those data entered into LoggerPro, we can now find velocity by copying the position column, paste it one cell lower and adjacent to the original column, and subtracting the two. Partial results for a run are shown in the table.

Once we divide the by 1/30^th of a second, we can plot velocity vs. time. You might wonder about the “noise” in the plot. Were we not careful enough when analyzing the video? It might be related to the times when the feet touch the floor and the not-so-smooth start to a run. (You should note the points of footsteps in the raw video to determine if your curve shows those). We will also find that there is inherent “noisiness” in these slope calculation curves because of how we do the calculations to get them. We take quantities like position from a ruler which we might measure to ½ mm resolution and divide them by a very short time interval. This process tends to amplify the fact that we do not measure with a perfectly fine precision and reveals the graininess of our data.

In order to find the acceleration, we could take one more slope calculation .

However we might expect even more “noise” in our plot. One way to estimate a is to do a linear fit to part of the data. We have shown what this looks like in LoggerPro. The slope of the velocity curve is acceleration. This approach allows us to estimate the acceleration of the dancer under varying conditions; for example using different starting positions.

Finally, we should be careful that our numbers make sense. The backdrop in the dance studio is about 3m long. You should therefore expect displacements of no more than about 3m depending on how you set up your camera. It is easy to see on the video software how much time elapses during the runs. If it takes 1-2 seconds to travel 3 meters, your velocity values should be a few meters per second. If you get wildly bigger or smaller results, suspect that you have made a mistake.