# Calculation of permissible error in physical quantity

Tags:
1. Mar 20, 2016

### Abhishek Gupta

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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.

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

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?

2. Mar 20, 2016

### Staff: Mentor

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.

3. Mar 20, 2016

### 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 .

4. Mar 20, 2016

### Staff: Mentor

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.

5. Mar 20, 2016

### Abhishek Gupta

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

6. Mar 20, 2016

Sure.