In the hints below, we intersperse discussion with actual sample data , following sig. fig rules every step of the way. The most important thing is to NOT round off until the last final step; keep the extra digits, but remember to label them as significant or not, usually with an underline. For example 45.98567132 suggests only 4 sig. figs. But we keep the digits to the right of the 8 to avoid round off errors in intermediate computations. |
DATA SHEET:
(Q-1) Here we assumed an
error of plus or minus 1 g or +/- 1 g. Suppose you have a data
set like below. The thing to remember is the number of significant digits
increases from 3 to 4 when you compute the average = (sum of numbers) /5 = 3150/5 = 630.0 since 3150 has 4 sig. fig. You choose the maximum of R/N and Δminst, which in this case is the latter. If you were reporting a final result you would write, m = mBEST + Δm = 630.0 g +/- 1g. Note we would consider mBEST to have only 3 sig. fig in this case. On the other hand, if we were to have used 0.8 g as the uncertainty, we would write 630.0 g +/- 0.8 g; clearly we are not doing that since the instrument error is larger, but it's good to see this possibility. More hints later . |
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MASS m (g) |
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mBEST |
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R/N |
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Δminst |
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Δm |
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(Q-1)
The radius r is a
continuation of the process of determining the dynamic centripetal force magnitude
F. Measured in m, r takes on the following values using a meter
stick to 4 sig. figs. Note that the last digit is either a 5
or a 0 since we are rounding to the nearest 0.10 cm/4 = 0.025 cm = 0.0025
m, which we will round to 0.03 cm = 0.0003 m in the final step. Now, when
I compute the average, I still get 4 sig figs since the sum of the numbers
is also 4 sig fig: (0.8220 m )/5 = 0.1644 m.
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RADIUS r (m) |
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rBEST |
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R/N |
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Δrinst |
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Δr |
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(Q-3) Bob mass M is a continuation of the process of determining the dynamic centripetal force magnitude F. Measured in g, M takes on the following values using a mass scale to 4 sig. figs. Note that the last digit is either a 5 or a 0 since we are rounding to the nearest 0.10 g/4 = 0.025 g, which we will round to 0.03 g m in the final step. Now, when I compute the average, I get 5 sig figs since the sum of the numbers is 5 sig fig: (2255.90/5 = 451.180 g. |
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MASS M (g) |
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MBEST |
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R/N |
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ΔMinst |
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ΔM |
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complete the data sheet based on the above discussion. Follow
the suggested rules.
(Q-4)
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PERIOD T (s) |
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0.687684 |
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0.689008 |
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0.682688 |
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0.686164 |
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0.688060 |
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0.685832 |
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0.682448 |
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0.684936 |
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0.692860 |
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0.683928 |
TBEST |
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R/N |
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ΔTinst |
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ΔT |
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(Q-5) and (Q-6)
Below, I will show you how to compute F = FBEST + ΔF.
First off, we compute FBEST (formula (3) previous
page)
then compute uncertainty under (Q-6), ΔF=
# Formula (3): FBEST = 4π2
*MBEST*rBEST/TBEST2
= 4π2 *(0.451180)*(0.1644)/(0.6863608000
)2 = 6.215865913
(N). Now, let's look at each term within
the parentheses. The first term is the smallest to be included since I have underlined the least sig.
fig. in all three cases and will now add them, realizing the least sig.
fig. is in the 0.000001 place:
Since the left hand is less than the right hand side, the
discrepancy test is true. We can say the centripetal force equals the
spring force keeping the mass moving in a circle.
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