Cubic vs. Square fit: the Free Fall of plastic ball


We perform cubic and square fits with the data of the free fall of the plastic ball.
The difference between the fits is given by the errors in s and t . A table summarizing the results is provided at the end.

Note: the propagation of the errors is done by combining the calibration error in s with the reading error in quadratures. While this is wrong in principle (calibration and reading errors have different statistics - one is systematic, the other one is random error) the effect of this wrong-doing is not that big. It works!
The correct approach would be to take the reading error of s to be the total error - for low values of s. For large enough values of s, the calibration error (guide sheets say it is s/4000 ) is bigger than the reading error, so you take the calibration error as your total error in s.
As a final note: combining the calibration and readings errors as I did might be not that wrong in this case for a good reason. Imagine that the calibration error is generated by a "random" stretching/squeezing of the meter tape during the manufacturing process. So, some regions of the tape are shorter than those of a "right" tape, while others are longer. Since we don't know a priori what portions of the tape are stretched / squeezed (and assuming that the lengthscale of the anomalies is short enough - say, 10cm or so), the expectation is that the calibration error works as a random error in this case. Hence, the quadrature combination of the two.

s (± Sqrt[2*(0.3)2+(s/4000)2] mm) vs. t (± 0.1 ms)

g2fit-1.gif (1-2)

 

v0 = (1.8068 ± 0.0026) m/s
g = (9.465 ± 0.011) m/s2

= 11.095 for 8 d.o.f.

( probability: 19.6377)

g3fit-1.gif  (1-3)

v0 = (1.7902 ± 0.0066) m/s
= ( 19 ± 7 ) 10-3 m-1
g = (9.70 ± 0.09) m/s2

 

 

 

= 3.53956 for 7 d.o.f.

( probability: 83.1018)

 

s (  ± Sqrt[2*(0.3)2+(s/4000)2] mm) vs. t ( 0.05 ms)

g2fit-2.gif  (2-2)

 

v0 = (1.8071 ± 0.0022) m/s
g = (9.464 ± 0.009) m/s2

= 16.7973 for 8 d.o.f

( probability: 3.22909)

g3fit-2.gif  (2-3)

v0 = (1.7896 ± 0.0056) m/s
= ( 19.6 ± 5.7 ) 10-3 m-1
g = (9.71 ± 0.06) m/s2

 

= 5.22946 for 7 d.o.f

( probability: 63.1983)

 

s (  ± Sqrt[2*(0.2)2+(s/4000)2] mm) vs. t (  ± 0.05ms)

g2fit-3.gif  (3-2)

 

v0 = (1.8060 ± 0.0018) m/s
g = (9.469 ± 0.008) m/s2

= 23.728 for 8 d.o.f

( probability: 0.254523)

g3fit-3.gif  (3-3)

 

v0 = (1.7905 ± 0.0043) m/s
= (18.6 ± 4.7) 10-3 m-1
g = (9.70 ± 0.03) m/s2

= 7.84475 for 7 d.o.f

( probability: 34.6482)

 

s (  ± Sqrt[2*(0.2)2] mm) vs. t (  ± 0.05ms)

g2fit-4.gif  (4-2)


v0 = (1.8083 ± 0.0014) m/s
g = (9.458 ± 0.006) m/s2

= 43.1285 for 8 d.o.f.
( prob: 0.0000830799)

g3fit-4.gif  (4-3)

v0 = (1.7895 ± 0.0037) m/s
= (19.8 ± 3.7) 10-3 m-1
g = (9.715 ± 0.038) m/s2

 

= 13.223 for 7 d.o.f .

( probability: 6.68484)

 

s (  ± Sqrt[2*(0.3)2] mm) vs. t (  ± 0.05 ms)

g2fit-5.gif  (5-2)


v0 = (1.809 ± 0.0018) m/s
g = (9.4556 ± 0.0074) m/s2

= 25.8416 for 8 d.o.f.

( probability: 0.11178)

g3fit-5.gif  (5-3)

 

v0 = (1.789 ± 0.005) m/s
= (20.1 ± 4.7) 10-3 m-1
g = (9.72 ± 0.05) m/s2

= 7.91985 for 7 d.o.f.

( probability: 33.9715)

 

s (  ± Sqrt[2*(0.3)2] mm) vs. t (  ± 0.1 ms)

g2fit-6.gif  (6-2)

 

v0 = (1.8078 ± 0.0024) m/s
g = (9.46 ± 0.01) m/s2

= 14.178 for 8 d.o.f.

( probability: 7.72419)

 

g3fit-6.gif  (6-3)

 

v0 = (1.7899 ± 0.0062) m/s
= (19.5 ± 6.3) 10-3 m-1
g = (9.71 ± 0.06) m/s2

= 4.39083 for 7 d.o.f.

( probability: 73.382)

To summarize, we display below the main results from the above plots (rounded to significant digits)

Plot #
and
fit type

v0

(m /s)

g

(m/s2)

(m-1)

x 103

reading
error in

s

(mm)

err. in

t

(ms)

/ d.o.f.
1-2
1.807 ± 0.003
9.465 ± 0.011
-
0.4
0.1
23.7 for 8
1-3
1.790 ± 0.007
9.70 ± 0.09
19 ± 7
0.4
0.1
3.53 for 7 .
2-2
1.807 ± 0.002
9.464 ± 0.009
-
0.4
0.05
16.8 for 8
2-3
1.790 ± 0.006
9.71 ± 0.06
19.6 ± 5.7
0.4
0.05
5.23 for 7
3-2
1.806 ± 0.002
9.469 ± 0.008
-
0.3
0.05
23.7 for 8
3-3
1.790 ± 0.004
9.70 ± 0.03
18.6 ± 4.7
0.3
0.05
7.84 for 7
4-2
1.8083 ± 0.0014
9.458 ± 0.006
-
0.3*
0.05
43.1 for 8
4-3
1.790 ± 0.004
9.715 ± 0.038
19.8 ± 3.7
0.3*
0.05
13.2 for 7
5-2
1.809 ± 0.002
9.456 ±0.007
-
0.4*
0.05
25.8 for 8
5-3
1.789 ± 0.005
9.72 ± 0.05
20 ± 5
0.4*
0.05
7.92 for 7
6-2
1.8078 ± 0.0024
9.46 ± 0.01
-
0.4*
0.1
14.2 for 8
6-3
1.790 ± 0.006
9.71 ± 0.06
19.5 ± 6.3
0.4*
0.1
4.39 for 7

* in those cases the error in s doesn't account for the (s/4000) calibration error. See the plots for details on error calculus!

Moral:

The value of g retrieved with a square fit (frictionless theory) is typically (9.46 ± 0.01) m/s2.
It is definitely far from the real value (g = 9.80m/s2, in MP building).
It looks like the square fit is not a good choice; this is also pointed out by the values (typically 15...25 for 8 d.o.f.).

We can see a clear improvement in the case of a cubic fit (theory with friction).
The value retrieved is typically g = (9.71 ± 0.06) m/s2 which, in the 2-sigma interval, "catches" the precisely measured g (= 9.80 m/s2).
values are typically 4...8 for 7 d.o.f., which also shows that the cubic fit is a good choice.

© Sorin Codoban, 2003.

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