Video Project Final Analysis

Over the course of this video project, we set out to analyze the real life application of two physics topics we have learned throughout the year in class. We analyzed both simple harmonic motion as well as projectile motion, with the projectile being Cam’s body. As Cam was on the swing, he was subject to simple harmonic motion, with the swing acting as a simple pendulum. From this motion, we withdrew several different aspects of the pendulum, including period, frequency, angular frequency, and amplitude. After completing half of a period of the pendulum’s motion, Cam then exited the swing and underwent projectile motion. From this motion, we withdrew even more information, including: x and y position, velocity, acceleration, and different forms of energy. We found through the course of this project that the techniques we learned in class still apply to real-world situations, yet with some error involved. It was particularly interesting to notice how all of the different information coincided with each other. For example, when potential energy was at a minimum in Cam’s motion, his kinetic energy was then at a maximum. Although this video analysis was not done in an ideal “physics” world, the data obtained from the video was still quite valid. Furthermore, the analysis we ended up obtaining from this data allowed us to analyze not just one form of motion- we were able to demonstrate both SHM and projectile motion.

Pendulum Motion Analysis

For the first part of the motion in this video, Cam’s body is moving on the swing in simple harmonic motion. Therefore, the system can be treated as a simple pendulum, with the swing having a length of 2.197 meters.  From the analysis of the y-position vs time graph for the time while on the swing, the amplitude drawn from the data is 0.67135 meters. Furthermore, the angular frequency drawn from this system is equal to the square root of gravity over the length, giving us an angular frequency of 0.473 radians/second. Cam’s body leaves the swing at the leftmost point in the swing’s motion, which is equal to half of the swing’s period. Cam leaves the swing at time t=1.235 seconds, so this provides that the swing’s total period is 2.47 seconds long. Since frequency is the inverse of the swing’s period, this leads us to obtain a frequency for the system of 0.405 Hz. Similarly, if angular frequency is multiplied by 2  the value for frequency is .337 Hz and the period found by the inverse of frequency is 2.972. Since the theoretical and the observed values for frequency and period are very close it shows the accuracy of our lab. With percent errors of 20.2% and 16.8% it is easy to see that the data was quite accurate.

List of Resources/Links

For further help obtaining the information of the pendulum in this system, we utilized the following websites:

·         http://www.physicsclassroom.com/Class/waves/u10l0c.cfm

·         http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html

Sample Calculations

For a lot of our calculations we used excel to produce graphs and lines of best fit to represent the data that we had collected. For example for the y-acceleration we found the linear line of best fit and compared our slope to the value of the accleration due to gravity. Addtionally other mass we calculated we found the kinetic and potential energy we used the equations K= 1/2mv^2 and U=mgh. We took half of Cam's mass, after converting it from pounds, and multiplied by the velocity vector squared to find the kinetic energy of the motion. And then we took Cam's mass and multiplied it by the acceleration due to gravity and Cam's y-position to find the potential energy of the system. Then to find the total energy we found the sum of these two values, and to get one value for the total energy we averaged all of our total mechanical energies. For the simple harmonic motion of the pendulum we had to calculate the amplitude, period, angular frequecy, and frequency of the motion. We found amplitude by using our y position vs. time graph to find the distance from the line of equilibrium. To find the period we used our x position vs. time graph to figure out the time at which Cam's body leaves the swing. This value is equal to half of the period which made it easy to find. For the angular frequency we plugged our data into the equation ω = √(g/l). And we found frequency by taking the inverse of period (1/T).

Total Mechanical Energy

For the total mechanical energy of our systen we found the average sum of the kinetic energy and the potential energy. We found that the mechanical energy hovered around 864.0717 Joules. Our calculated energy does not stay perfectly constant probably due to friction in the swing and air resistance.

Kinetic Energy

Here is our graph of Kinetic Energy versus time here using our values of velocity in the x and y direction I used vector addition to find the resulting velocity which I then squared. After obtaining v² I multiplied it by Cam’s mass and then divided it by two. These steps gave me the kinetic energy using 1/2mv² at each time. You can see that the kinetic energy peaks between .5 and 1 seconds and between 1.5 and 2 seconds. Which means these are points where the velocity is the highest.

Potential Energy

Here is the graph of Cam’s potential energy versus time. As you can see the potential energy graph shows peaks and dips over time. At the beginning of Cam’s motion where he is at the top of the swing’s period the potential energy is at its highest. Also the potential energy peaks again when Cam leaves the swing at the other end of the swing’s period. Another interesting note to keep in mind is that energy is conserved. When comparing the two energy graphs you can see that the peaks of the kinetic energy graph are also the dips of the potential energy graph.

Y Velocity in the air

Here is the graph of the y velocity versus time after Cam has left the swing. Here the data also represents an object in projectile motion. The velocity is shown decreasing at a pretty consistent rate. Using a linear trendline we can see that indeed the data is sloping downwards. The slope of this function gives us the constant acceleration the object is experiencing. The slope of our graph is -9.748 which is extremely close to the accepted value of acceleration due to gravity which is -9.810. This makes sense because an object in projectile motion experiences a constant acceleration due to gravity which is demonstrated in our data. The slight difference between our value of acceleration and the accepted one could be due to air resistance affecting Cam’s acceleration in the y-direction.

Y Velocity on the swing

The graph above is Cam’s v velocity versus time while on the swing. This graph is a very good representation of his y velocity while on this swing. As you can see it looks very similar to the position at this time that is because for the function is the derivative of the position function which makes it the opposite of the position function. For example if that was a sine function the velocity would show a similar looking cosine function. This also demonstrates a changing acceleration in the y direction while on the swing. This makes sense as well because in simple harmonic motion ion the acceleration is constantly changing because the direction is constantly changing.

x Velocity

The velocity in the x direction matches the motion of the object. The object in the beginning of the motion has a varying x acceleration because it is in simple harmonic motion and changing direction. However when Cam leaves the swing the acceleration becomes more constant at about -2 m/s². Although an ideal object in projectile motion would have an x acceleration of 0 we must take in to account air resistance with is likely creating a negative acceleration and slowing Cam down in the x direction.

Y Position in the air

This a graph of Cam’s position in the y-direction versus time. Here you can see how his motion as he leaves the swing resembles a projectile motion graph. Where his position is arced downward due to the acceleration due to gravity. Also you can see that the function we chose to fit this graph is a quadratics function which is the same that an object in projectile motion would be. The r² value of .9901 shows a strong correlation between our data and the function.

Y Position on the Swing

This is a graph of Cam’s position in the y direction while he is still on the swing. The function that models the graph is a fourth order polynomial function. This graph is a Taylor polynomial function for a trigonometric sine or cosine graph. Either one of these functions would make sense because the motion on the swing is similar to a pendulum, with an amplitude and period.

X Position

Here is our graph of the position in the x-direction versus time. This describes Cam’s motion in the x-direction as he swings and jumps off the swing. As you can see the x direction while on the swing (the first half of this graph shows a slightly curved pattern which emulates the motion of a pendulum. However when Cam jumps off the swing his motion represents that of an object in projectile motion. Therefore, his x position shows a linear line from where he leaves the swing to where he lands on the ground.

Table of Values

Here is a table of our raw data. We gathered this data by using the VideoPoint program. All of the anaylsis we did came from using these values and data.

Frame	Time [s]	x position [m]	y position [m]	x velocity [m/s]	y velocity[m/s]
1	0	4.2759	1.1512	4.3022	1.138
2	0.101	4.2496	1.1643	-0.1283	-0.032
3	0.134	4.2233	1.138	-0.897	-0.3986
4	0.167	4.1904	1.138	-0.7081	-0.8092
5	0.201	4.1773	1.0854	-0.4764	-0.8576
6	0.234	4.1575	1.0788	-0.7973	-1.6944
7	0.267	4.1246	0.9736	-1.315	-3.0348
8	0.301	4.072	0.8815	-0.8576	-1.3341
9	0.334	4.0654	0.8815	-0.6977	-1.3954
10	0.367	4.0259	0.7894	-1.8208	-2.9336
11	0.401	3.947	0.6907	-2.4777	-2.7636
12	0.434	3.8549	0.5986	-2.8325	-2.7313
13	0.468	3.7628	0.5131	-1.5247	-1.0482
14	0.501	3.7497	0.5262	-1.8937	-0.897
15	0.534	3.6378	0.4539	-3.8441	-2.4278
16	0.568	3.4997	0.3683	-4.0977	-2.2871
17	0.601	3.3549	0.296	-4.6846	-1.8937
18	0.634	3.1905	0.2434	-2.7313	-0.6069
19	0.668	3.1773	0.2565	-2.2871	-0.5717
20	0.701	3.0326	0.2039	-5.2826	-0.9967
21	0.734	2.8287	0.1907	-5.7661	0
22	0.768	2.6576	0.2039	-4.5742	0.2858
23	0.801	2.5129	0.2105	-2.529	0.1011
24	0.835	2.4932	0.2105	-2.1918	0.4764
25	0.868	2.3616	0.2434	-4.5849	1.0964
26	0.901	2.1906	0.2828	-4.6533	1.6185
27	0.935	2.059	0.3486	-3.7166	2.1918
28	0.968	1.934	0.4341	-2.3921	1.5947
29	1.001	1.9011	0.4539	-2.1243	1.2139
30	1.035	1.7959	0.5131	-3.0495	1.8106
31	1.068	1.6906	0.5789	-3.0348	1.922
32	1.102	1.5985	0.6381	-2.8589	2.0965
33	1.135	1.4933	0.7236	-1.5947	1.2957
34	1.168	1.4933	0.7236	-1.6185	1.315
35	1.202	1.388	0.8091	-3.1448	2.0965
36	1.235	1.2762	0.8683	-3.2892	1.0964
37	1.268	1.1709	0.8815	-2.8325	0.3034
38	1.302	1.092	0.888	-1.5247	-0.1905
39	1.335	1.0657	0.8683	-1.7941	-0.0996
40	1.368	0.9736	0.8815	-2.529	0.3034
41	1.402	0.9012	0.888	-2.3824	-0.2858
42	1.435	0.8091	0.8617	-2.7313	-1.1127
43	1.469	0.7236	0.8157	-1.5247	-0.4764
44	1.502	0.7038	0.8288	-1.3954	-0.7973
45	1.535	0.6315	0.763	-2.7313	-2.2255
46	1.569	0.5262	0.6841	-2.6683	-2.573
47	1.602	0.4473	0.5854	-2.4918	-2.9901
48	1.635	0.3618	0.4868	-1.6185	-1.4162
49	1.669	0.342	0.4933	-2.0012	-1.8106
50	1.702	0.2236	0.3618	-3.1895	-4.0865
51	1.735	0.1315	0.2236	-2.3267	-3.5406
52	1.769	0.0723	0.1315	-1.9059	-2.2871
53	1.802	0	0.0657	-1.315	-0.8092
54	1.836	-0.0131	0.0789	-0.7623	-0.0952
55	1.869	-0.0526	0.0592	-0.2096	0.6186