course Phy 201
I'm having trouble seeing how some of the values used in your sample calculations relate to your data. See my two notes and see if you can clarify.
Let me know if there's anything you need me to explain further.
Phy 201Experiment #24: The acceleration of an object moving in a circle at constant velocity is v^
Data from experiment:
Length of string (m) 1m 0.8m 0.6m 0.4m 0.2m
Distance (m) 3.3 2.49 2.29 1.47 1.04
3.4 2.72 2.54 1.55 1.22
3.66 3 2.57 1.7 1.37
Total Height of Mass (m) Final Velocity (m/s) dt (s)
2.27 6.6702324 0.68063596
2.07 6.3696154 0.64996075
1.87 6.0540895 0.61776424
1.67 5.7211887 0.58379476
1.47 5.3676811 0.54772256
I used the equations vf^2 = v0^2 + 2a’ds in order to get the final velocity and then ‘dt = ‘ds / vAve for the time it took the object to fall. Then by multiplying ‘dt by the horizontal distance, I find the horizontal velocity of the mass.
I assumed that v0 = 0m/s (I’m not sure if this is right or not), a = 9.8m/s^2, ‘ds = the height of the weight.
For example, the 1meter length has a final velocity of:
vf^2 = v0^2 + 2a’ds
vf = sqrt (2 * 9.8m/s^2 * (1.27m + 1m))
vf = 6.67m/s
So for the time interval at 1meter,
‘dt = ‘ds / vAve
‘dt = 2.27m / (6.67m/s / 2) = 0.68s
For the horizontal velocity, I multiplied the time it took for the object to fall by the distances the mass went in the horizontal direction (which are given in the very first table).
For example: at the 1meter height, ‘ds / ‘dt for the first test is 3.3m / 0.68s = 4.85m/s
Horizontal Velocity (m/s) 1m 0.8 0.6 0.4 0.2
4.848 3.83 3.706 2.518 1.899
4.995 4.185 4.111 2.655 2.227
5.377 4.615 4.160 2.912 2.501
I only graphed the second set of data that I took from the previous graph. I couldn’t get the graph to work well using all three tests for each radius.
Acceleration at 1m Acceleration at 0.8m Acceleration at 0.6m Acceleration at 0.4m Acceleration at 0.2m
5.044959184 3.274032398 3.335532993 1.841175 1.6224
5.355346939 3.906808163 4.103568707 2.047028061 2.2326
6.205716735 4.75255102 4.20107585 2.462397959 2.81535
The equation for acceleration is a = v^2 / r, so for the 1m height, I calculated:
a = (2.25m/s) ^2 / 1m = 5.04m/s^2
In your sample calculation you use 2.25 m/s as velocity. However I don't see that velocity represented on the table, and it doesn't seem to correspond to any of the quantities related to your 1 meter radius.
In order to best linearize my data set; I used a power of 0.5. This is the table of the radius and v^p:
Radius (m) Vel. With Power v^.5
1 1.521237079
0.8 1.329621469
0.6 1.252645664
0.4 0.951252795
0.2 0.817448176
Again I'm having trouble relating the v^.5 values to the v values you report.
This graph has a slope of 0.951.
1/k = v^p / r
2.235023086
2.557159585
3.379525407
4.073557139
7.46226418
I believe that my results here do not validate the centripetal acceleration, a = 9.8m/s^2. I redid my calculations for the velocity twice now and am hoping that I did get the velocity and acceleration correct, but if this 1/k value is supposed to be close to 9.8m/s^2, I’ve still got something wrong with my data.
There is quite a bit of room for experimental error in this experiment. As mentioned in the instructions, the point of release will never by exact each time the weight is released and a person controlling the speed of the weight will also vary significantly with each length. Measuring the distance is probably less important than the other variables because the numbers tend to be so high for the distance that an inch or two in variation may not cause that many problems in the results. The main problem in the distance measuring is deciding where exactly the weight first hit the ground.
I think that the maximum likely error would be the angular position of the mass at the instant of the release because releasing the weight at the exact same angle each and every time the object is thrown is highly unlikely for a person to be able to do.
Though I’m not sure if this is what the question is asking for, I think the maximum likely angular distance from vertical, in degrees of arc, at which the string went slack was approximately 225 degrees. The more slack in the string, the lower the altitude of the weight and the shorter distance the object would go. If we spun the object as fast as possible, the altitude would increase when the object was released and would travel much further.
I think the maximum likely error in my determination of the horizontal range of the projectile was the fact that I let a 3 year old do my measurements for me. Also, I used his toys to mark the spots where the mass landed and sometimes, they would get shifted around or kicked. Given my results, I feel my friend Andrew did rather well in his measurements, especially for a kid his age. I’m not sure the horizontal range would have that much of an affect on the velocity of the object for the 1m length. I think the shorter lengths would be affected more by a variation in the horizontal range.
I stuck my arm straight out at shoulder height to release the object and I think that this vertical altitude was pretty equal throughout all the tests. To get the mass spinning, my arm did rotate some but I feel that overall, I was able to keep it at an accurate height of 1.27m. If I’d lowered my arm significantly during the 1-meter trial, the velocity would probably be lowered slightly. I’m not entirely sure though.
Order of errors from least to most important:
1. The vertical altitude from which the mass was released. If I’m correct in assuming that this is the height of my arm when I released the object combined with the length of the string, I feel choosing to keep my arm straight out and perpendicular to my body was the best idea and helped keep this error from fluctuating too much.
2. The horizontal range measurement is probably the next most problematic error. It’s fairly easy to measure the distances correctly but remaining accurate in where the mass initially landed can cause a little fluctuation. I still think that this can be better controlled than the moment of release and the speed of the object.
3. The angle that the object was released would be my third choice in which errors are most important. I felt like I was able to release the objects at a fairly equal time, and tried to let it be the very top of the arc, but I know that sometimes I released too early and sometimes too late. Still, the variability in what range the object was released can only be about 45 degrees (with straight up being the center), otherwise I would have noticed that I’d let go of the mass too soon or too late.
4. The most important error I feel is the point at which the string went slack. This speed, especially for me, was very hard to obtain, much less remain equal throughout all the tests. I think that this directly affects the distance of the object and know that for a fact because at on point, the mass was going really slowly and I accidentally let go of the weight. I still let go at about the right time, but the object hardly went half the distance of my previous tests with that length.
As for the part in the instructions that says to “place bounds on the uncertainty in your determination of the horizontal range of the mass, and determine the resulting range of the values of 1/k in the above analysis”, I’m not entirely sure what this (or the following questions) mean. I think I may have done the 1/k = v^p / r wrong because I haven’t used the horizontal ranges in any of the calculations, only the radius, acceleration and velocity.
I didn’t go as far as designing an experiment using a machete or a blowtorch (though I admire the mind who did), I just thought of ideas that would help minimize the errors in this experiment. Such as using a machine to spin the mass, this way the height is stable and the speed can be set or altered if needed. Don’t allow a 3 yr old to help with the measuring aspect and do the experiment indoors so the wind and ground slope outside do not affect the object.
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