The Effects of Air Resistance When Trying to Calculate a Value for ‘G’
Stefan Ruttimann
ID:110708680
Lab Partner: Dylan Thompson
PC 131, Lab 5
Lab Instructor: Terry Sturtevant
Lab IA: Susan Shaw
October 25, 2011
Purpose: The purpose of this experiment is to recognize what effect air resistance has, if any, on an object in free fall. By using various types of objects and dropping them at a height, we can see what effect air resistance has on the different objects by analysing the measured times and comparing them. Quantitatively, we will be able to calculate the acceleration of gravity using our times and height in accordance with our formula for g. As a qualitative goal, we can compare our calculation of g with the standard value for gravity and assess what effect wind resistance had on the object.
Method: Refer to Lab Manual, Chapter 14
Change in Method: For one of our trials we had completely inaccurate times that were measured because of our timing approach so we redid the trial with a new approach and used the second trial for our numbers.
Experimental Results:
Instrument Used:
Measuring tape Precision Measure is 0.0005m
Stop Watch Precision Measure is 0.01 s
Raw Data:
Before we did any timing we had to measure the distance between the dropping height and the bottom of the bucket and we found it to be 5.39±0.02m. There are 5 different trials that we had to do to see the qualitative effects of air resistance. They are as follow:
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1. Person A drops the ball and person B times.
2. Person A drops the ball and person A times.
3. Person B drops the ball and person B times.
4. Person B drops the ball and person A times.
5. Person, either A or B drops ball two and person, either A or B times.( In our case person A dropped and person A timed)
The most important thing about doing these trials was beginning and ending the timers correctly. If the timing was off then when we go do our calculations our value for ‘g’ will be off according to how we timed the dropping. The precision measure of the stop watch is 0.01s and the uncertainty in the time is about ±0.02. A slight increase in the height of the dropped ball would be from holding the ball a bit lower or higher from where it was actually measured. Similarly, differences in time can be seen. If a timer starts the watch too early or ends it too late the time would be a bit longer then if the timer were to time it properly. Also if the timer starts it too late and ends it a bit too early then the time would be a bit shorter than if the timer were to time it properly. Table 1 presents the raw data measured during the lab. Table 2 is the uncertainties
Stop Watch |
units | Seconds |
pm | 0.01s |
Ball | Ping Pong Ball | Steel Ball |
Dropper | A Dropping | B Dropping | A Dropping |
Timer | Gofer B | Dropper A | Dropper B | Gofer A | Dropper A |
1 | 0.82s | 0.9s | 1.09s | 1.1s | 1.05s |
2 | 0.8s | 1s | 1.04s | 1s | 1.02s |
3 | 0.87s | 1s | 1.16s | 1.25s | 1.06s |
4 | 1.02s | 0.94s | 1.03s | 0.9s | 1.06s |
5 | 0.85s | 0.97s | 1.03s | 1.16s | 1.18s |
Table1: Times for Trials
Person A: Stefan Ruttimann (myself)
Person B: Dylan Thompson
Height: 5.39 ±0.02m
Symbol | factor | Bound | Units |
h | Bend in tape measure | ±0.01 | m |
h | Placement of hand | ±0.01 | m |
t | Stopwatch begin | ±0.01 | s |
t | Stopwatch end | ±0.01 | s |
Total h | Total | ±0.02 | m |
Total t | Total | ±0.02 | s |
Table 2: Uncertainty in h and t
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Stop Watch |
units | Seconds |
pm | 0.01s |
Ball | Ping Pong Ball | Steel Ball |
Dropper | A Dropping | B Dropping | A Dropping |
Timer | Gofer B | Dropper A | Dropper B | Gofer A | Dropper A |
Average | 0.872s | 0.962s | 1.07s | 1.082s | 1.06s |
Average Rounded | 0.9s | 1s | 1s | 1s | 1s |
standard deviation of Time | 0.09s | 0.04s | 0.06s | 0.1s | 0.07s |
Standard Deviation of the Mean | 0.04s | 0.02s | 0.03s | 0.06s | 0.03s |
Uncertainty in Time | 0.04s | 0.02s | 0.03s | 0.06s | 0.03s |
Table 3: Calculations and Uncertainty
Person A: Stefan Ruttimann (myself)
Person B: Dylan Thompson
Height: 5.39 ±0.02m
Stop Watch |
units | Seconds |
Ball | Ping Pong Ball | Steel Ball |
Dropper | A Dropping | B Dropping | A Dropping |
Timer | Gofer B | Dropper A | Dropper B | Gofer A | Dropper A |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
Table 4: G Calculations
Trial | | Percent Difference |
DA-GA | 0.12≰ 0.08 | 11.1% |
DB-GB | 0.198 ≰0.07 | 18.5% |
DA-GB | 0.09≰ 0.06 | 9.4% |
DB-GA | 0.01≤ 0.09 | |
Table 5: Uncertainty Agreement
| Gofer B | Dropper A | Dropper B | Gofer A | Dropper A |
Optimal Trials | 16 | 4 | 9 | 36 | 9 |
Table 6: Optimal Trials
Instrument |
Name(or reference) | Circuit Board Fixed Intervals of 500msAnticipated Data |
Units | milliseconds |
Precision measure | 0.001s |
Triali | Person ADylan Thompson | Person BStefan Ruttimann(myself) |
12345 | 214ms203ms71ms16ms159ms | 16ms5ms38ms38ms27ms |
MinMax | 16ms214ms | 5ms38ms |
t | 132.6ms | 24.8ms |
σ | 86.12ms | 14.34ms |
α | 38.51ms | 6.41ms |
∆(t ) | 40ms | 10ms |
Table 7: Anticipated event data
Instrument |
Name(or reference) | JavaSynchronization Data |
Units | milliseconds |
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Precision measure | 0.001s |
Triali | Person ADylan Thompson | Person BStefan Ruttimann (myself) |
12345 | 58ms51ms27ms105ms18ms | 27ms29ms8ms121ms21ms |
MinMax | 18ms105ms | 21ms121ms |
t | 51.8ms | 41.2ms |
σ | 34ms | 45.4 |
α | 15.2 | 20.29 |
∆(t ) | 20 | 20 |
Table 8: Synchronization data
Sample Calculations:
Sample Average Calculation:
Calculations for Standard Deviation:
| Times | | |
| 0.82 s | – 0.05 | 0.0025 |
| 0.80 s | – 0.07 | 0.0049 |
| 0.87 s | 0.00 | 0.0000 |
| 1.02 s | 0.15 | 0.0225 |
| 0.85 s | – 0.02 | 0.0004 |
Average time | 0.87 s | | |
| | | 0.0303 |
Table 3: Calculations for Standard Deviation
the standard deviation was 0.09 s and the uncertainty in time was 0.04 s.
Gravity:
We know that the value for g should be somewhere close to 9.81m/s2 [1]. We also have the height equation from the lab manual. [2] It is as follows.
If we rearrange this formula we get:
Sample for Dropper A and Gofer B:
By analysis we can plainly see that g is way too high of a value. This can be attributed to the fact that since t is smaller we know that g will be bigger. This means that timer and either waited to start the timer or ended the timer too soon, or a combination of both.
Uncertainty in g:
Δg by inspection is
Δg by algbra is
For Dropper A, Timer B:
Inspection:
Algebra:
Examples of Agreement of uncertainties:
Percent Difference Example:
Percent difference
=2.24%
Optimal Trials Calculation:
Summarized Calculation Results:
The most accurate time was with dropper A and Timer A. This is because the values were the most consistent. However, the ping pong ball trial produced a value of g closest to 9.81. If a person decides to delay the watch after the ball is dropped the average time will go down and therefore, value of g will be higher. If a person on the other hand delays the stop watch after it hits the ground the time average time will increase and therefore the value calculated for g will go down. If a person delays starting the watch after the ball is dropped and delays stopping the watch after the ball hits the ground by the same amount, the average time will remain the same and therefore, the value calculated for g will remain the same.
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Discussion of Uncertainties
Looking at table 5, we see that for DA-GA, DB-GB, and DA-GB their trials are 0.12≰ 0.08, 0.198 ≰0.07, and 0.09≰ 0.06, respectively. All three of these comparisons do not agree with experimental uncertainty. However, for DB-GA the comparison is 0.01≤ 0.09, and that does agree with experimental uncertainty. Since most of the trials do not agree the values seem to be significantly different. Q1 However, if we compare DA for the steel ball and DB for the ping pong ball, which were our two best trials according to their closeness to actual g, their uncertainties are 1.07±0.03 and 1.06 ±0.03 (see calculations section).
These do agree with experimental uncertainty which indicates that even though the trials for the ping pong ball didn’t all agree, the fact that the two different methods do proves that they are not significantly different. Q4 If we look at my calculation for the standard deviation of the mean calculation, it equalled 0.04s, and therefore it is bigger than the precision measure of the stopwatch of 0.01s. Since the standard deviation of the mean is the larger of the standard deviation of the mean and the precision measure (0.04), having a more accurate stop watch would not reduce this uncertainty .Q2 If we look at the times for our first trial with gofer B in Table 1, we see that the times are quite sporadic. Also, we see that in Table 6, the optimal amount of trials is 16. This is feasible because, since we might not have been accustomed to the experiment on the first trial, most of the successive trials fared better. In fact, the next trial with dropper A has optimal trials of only 4.Q3 Measurement of reaction times between gofers and droppers depends on how you do the experiment. If the dropper is to count down when he/she is going to drop the ball, then the gofer’s measurement would be anticipated because he/she would be able to know when the ball is about to drop. The landing of the ball is completely anticipated because both the dropper and the gofer can watch the ball as it is about to land. When the dropper releases the ball, he or she tries to drop the ball and start the stopwatch at exactly the same time and that measurement of reaction time would be synchronization. There is a direct correlation between how on time your counting is and the measured times. When we had sporadic counting, then the times would have much bigger ranges because you might start the timer earlier or sooner based on that fact.Q5 The precision measure of the tape is 0.0005 m. However, the realistic uncertainties are 0.001m and 0.003m. Since, in this case, the uncertainty is the realistic uncertainty and not the precision measure then a more accurate tape measure would not reduce the uncertainty in g.Q6
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Our preferred trial was actually with the steel ball and having dropper A time. Taking at a look at our calculations page, our uncertainty in g for our best trial does agree with experimental uncertainty because |9.81 – 9.51|≤0.00 + 0.6. We know that gravity does not have an uncertainty. The preferred trial with the ping pong ball was with dropper B timing. Again in the calculations section, we see that and this does agree with experimental uncertainty. Since both of the preferred trial for the two methods agree with their uncertainties no calculation of percent difference is required. Q7 For gofer B and dropper A, the values for g are higher than 9.81. This is because the average time is slightly lower than normal. This could be attributed to a timer starting the stop watch after it is released or ending a stop watch before the ball hits the floor. It also could be what the position of the droppers wrist was as he/she was about to drop the ball. If it was below what was measured, the times would be smaller resulting in a higher calculated g. Other things like the angle of the tape measure and whether you measures to the bucket or the float could also change the times. If the tape measure was angled then the measurement for h would be greater and that would result in a higher g calculation. The same thing applies to whether you measure to the bucket or floor. Taking a look at tables 7 and 8 we can see that for the anticipated data the bounds would be 0.132±0.04s for Dylan and 0.024±0.01 for myself. In the synchronization data the bounds would be 0.052±0.02s for Dylan and 0.041±0.02s for myself. The bounds would a bit lower because the times in those trials are smaller. Q8 Comparing gravity from the steel ball to the preferred ping pong ball trial (dropper B) with their uncertainties, we can see that they do agree with experimental uncertainty because 0.18≤1.2. Since the values do agree we can safely conclude that friction is not significant. Wind resistance seems to have almost no effect on the steel ball or the ping pong ball.Q9
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Conclusion:
I have concluded that the effects of wind resistance on a ball bearing are quite negligible.
While air resistance is always present and doing work on an object, in our experiment it did not show to have that much of an effect on our calculations. I believe that the effects of stopping and starting the stop watches had much more of an effect on the times and were the main reason our calculations might have been off. For example, when person a dropped the ball bearing and timed it, we measured g to be around 14.18m/s2. When we compare that to the ping pong ball trial with a calculated g of around 9.59m/s2, with the fact that we know the ping pong ball has lower density and greater surface area, we can realize that it had to be our error in timing. Also, in all of our trials the Δgt was more significant than Δgh. This proves that most of the uncertainty is based on our timing measurements and not our height measurements. If we were to increase the height of the dropping length for 5.39m to 100m, I believe that we would see results that would show the effects of wind resistance more clearly. To make this experiment more precise and consistent we could have a holder for the ball that activates a sensor to start a clock and when it hits the floor we could have another sensor that stops the clock and this would reduce the inaccuracy of the measurement of reaction times and the uncertainty in the height. Also we could acquire a beeper that counts to three so that on 3 we could drop it and therefore it would be more anticipated. One more thing we could do is increase the height of the dropping length so we have more time to watch the ball fall and anticipate its landing. Looking at dropper B with the ping pong ball which was the preferred trial, we see that |9.41-9.81|≤ 0.6 + 0.0. Therefore, this does agree with experimental uncertainty. This shows us that wind resistance isn’t significant enough to cause the values to not agree. Gravity of the steel ball compared to the preferred ping pong ball trial (dropper B) and their uncertainties shows us that they do agree with experimental uncertainty since 0.18 is less than1.2. We must not calculate the percent difference because they do agree. Again, since the values do agree it’s safe to conclude that wind resistance on either ball is not significant.
References
[1] – Fundamentals of Physics 9th edition by Jearl Walker
[2] – Terry Sturtevant. Pc131 lab manual. From PC131 web page, 2011.
