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Evaluation of the error determining the moment of inertia of the disc

A disc K of a mass m , inner radius r and outer radius R

The moment of inertia I of a disc K is determining by equation:

I m r 2^R 2^+ (2 / * :

so I is a function I=f(m,r,R) and an error ΔI=f(Δm,Δr,ΔR)

Measurement results shown in the table, quantity

n 10 :

r R

data 6.2 102 -^* m * 93.4 102 -^* m * 6.5 102 -^* m * 93.6 102 -^* m * 6.1 102 -^* m * 93.7 102 -^* m * 6.2 102 -^* m * 93.5 102 -^* m * 6.2 102 -^* m * 93.8 102 -^* m * 6.4 102 -^* m * 93.6 102 -^* m * 6.3 102 -^* m * 93.4 102 -^* m * 6.4 102 -^* m * 93.5 102 -^* m * 6.4 102 -^* m * 93.6 102 -^* m * 6.5 102 -^* m * 93.7 102 -^* m * 102 mat :

R

r

We measure the mass:

m 0.4 kg * :

Next we estimate the accuracy of the mass measurement based

on the accuracy of the measuring device

Δm 0.0005 kg * :

We calculate the arithmetic means of the radius r and R:

n data rows :

r data 1 col :

R data 2 col :

rs data 1 col 10 / (sum : 0.0632 m *

r data 1 col :

rs r sum n / : 0.0632 m *

R data 2 col :

RS data 2 col 10 / (sum : 0.9358 m *

RS R sum n / : 0.9358 m *

We calculate the average square errors of arithmetic means:

Δr r i el rs - (2^i 1 n sum n n 1 - (* / sqrt : 0.000442 m *

Δr r rs - (2^n n 1 - (* / sqrt : 0.0004 m *

ΔR R i el RS - (2^i 1 n sum n n 1 - (* / sqrt : 0.000416 m *

ΔR R RS - (2^n n 1 - (* / sqrt : 0.0004 m *

Now we calculate the moment of inertia:

I m rs 2^RS 2^+ (2 / * : 0.175943 kg * m 2^*

We calculate the error with which the moment of inertia was determined:

ΔI I x k el diff (Δx k el * k 13 sum sqrt :

where k is quantity of variables r,R,m , and Δx appropriate errors determined.

ΔI r 2^R 2^+ (2 / (2^Δm 2^* m r * (2^Δr 2^* + m R * (2^ΔR 2^* + sqrt : 0.002254 m 2^* kg *

Final result is:

I ΔI ± 0.178197 m 2^* kg * 0.173689 m 2^* kg * 21 sys

The above method is based on normal distribution, which is generally applied to

very large number of measurements n-> ∞.

With a small number of measurements, eg. n = 10, it should be used

make a t-test of Student-Fisher and correct the error account.