Significant figures in permissible error calculation

[Abhishek Gupta]

Homework Statement

I have doubt in calculating the permissible error. It goes as follows

Measure of two quantities along with the precision of respective measuring instrument is

A = 25.0 ± 0.5 m/s, B = 0.10 ± 0.01 s. A physical quantity C is calculated as C = A × B. What will be the value of C along with permissible error

Homework Equations

$\frac { ΔC } {C} = \Big ( {\frac { ΔA } {A} + \frac {Δ B} {B} } \Big )<br />$

The Attempt at a Solution

STEP 1.

In the literature it is clearly mention that number of significant figures in result C is governed by the following rule.

"In multiplication or division, the final result should retain as many significant figures as are there in the original number with smallest number of significant figures."

Going by this rule C= 25.0 x 0.10 = 2.50 m = 2.5 m (rounding off to two significant figures).
STEP 2.

$\frac { ΔC } {C} = \Big ( {\frac { ΔA } {A} + \frac {Δ B} {B} } \Big ) = \Big ( {\frac { 0.5 } {25.0} + \frac {Δ0.01} {0.10} } \Big ) =<br /> <br /> 0.02 + 0.1 = 0.12<br />$

ΔC = 0.12 × 2.5 =0.30 m

However, to what the significant figures after rounding off should the permissible error ΔC be reported. Should ΔC=0.30m or 0.3m or something else What is the rule governing this?

 

[mfb]

However, to what the significant figures after rounding off should the permissible error ΔC be reported. Should ΔC=0.30m or 0.3m or something else What is the rule governing this?

Don't give more than two digits on the uncertainty, and those only if you believe the second digit could make sense. Your dominant uncertainty is not given better than 1 significant figure (and that digit is a 1), so 0.3 m is appropriate.

 

[Abhishek Gupta]

Thanks a lot for a prompt reply!
So the governing rule is that uncertainty in the measurement should be reported to one significant figure .

 

[mfb]

Depends on the situation.

If your values would have been given as A = 25.00 ± 0.50 m/s, B = 0.100 ± 0.010 s or even B = 1.100 ± 0.080 s, I would give two significant figures for the uncertainties on the product.

 

[Abhishek Gupta]

Depends on the situation.

If your values would have been given as A = 25.00 ± 0.50 m/s, B = 0.100 ± 0.010 s or even B = 1.100 ± 0.080 s, I would give two significant figures for the uncertainties on the product.

So you mean to say that it depends upon the significant figures present in the error involved in measuring the dependent physical quantities.

 

[mfb]

Sure.