Why Is the Constant Excluded in Uncertainty Calculations?

[elitewarr]

For a question like In an experiment, the external diameter D and internal diameter d of a metal tube were found to be (64 +/- 2) mm and (47 +/- 1) mm respectively. What is the maximum percentage error for the cross-sectional area of the metal tube?

I will need to find the external area.
So, area = pi(1/2d)^2
But I'm confused over whether should the uncertainty change?
What I mean was 1/2(64 +/- 2) = (32 +/- 1) ??
Or will the uncertainty remain at 2? If it remains at 2, the percentage uncertainty will definitely change.

And why does the formula R=kAB, k is a constant and A and B are physical terms, has the uncertainty formula tR / R = tA / A + tB / B
where tR / R, tA / A, tB / B are fractional uncertainty.
Why is the constant excluded?
R = kAB = AB + AB + AB ... + AB
So won't the uncertainty add up?

Thanks.

 

[cortiver]

But I'm confused over whether should the uncertainty change?
What I mean was 1/2(64 +/- 2) = (32 +/- 1)

Yes. X = 64 +/- 2 means we are sure that X is between 62 and 66, which is equivalent to saying X/2 is between 31 and 33, hence X = 32 +/- 1. Multiplication by a constant always preserves relative error.

And why does the formula R=kAB, k is a constant and A and B are physical terms, has the uncertainty formula tR / R = tA / A + tB / B
where tR / R, tA / A, tB / B are fractional uncertainty.
Why is the constant excluded?
R = kAB = AB + AB + AB ... + AB
So won't the uncertainty add up?

The absolute uncertainty adds up. The relative uncertainty is unchanged since
$$\frac{\Delta R}{R} = \frac{k \Delta(AB)}{k AB} = \frac{\Delta(AB)}{AB}$$

 

[elitewarr]

Ok. Thanks for clearing up things.

"Multiplication by a constant always preserves relative error"

This is also due to the constant being canceled right?

 

[cortiver]

Ok. Thanks for clearing up things.

"Multiplication by a constant always preserves relative error"

This is also due to the constant being canceled right?

Yes, that's right.

 

[pmacias]

The use of constants in scientific formulas is necessary to ensure accuracy and consistency in measurements and calculations. Constants are values that do not change and are known with a high degree of confidence. In the case of the formula R=kAB, k is a constant that represents the relationship between the physical terms A and B. Without this constant, the formula would not accurately represent the relationship between A and B.

In regards to the question about finding the maximum percentage error for the cross-sectional area of the metal tube, it is important to note that the uncertainty in the measurements of D and d will affect the uncertainty in the calculated area. However, the use of the constant pi in the formula for area ensures that the uncertainty in the diameter measurements will not directly affect the uncertainty in the calculated area. Instead, the uncertainty in the diameter measurements will be accounted for in the overall uncertainty of the calculated area.

To answer the specific question about whether the uncertainty in 1/2d should change to 1, it is important to understand that the uncertainty in a measurement is typically expressed as a range of values, rather than a single value. This is why the uncertainty in 1/2d is still expressed as 2, even though the value of d is known with more precision. This is also why the uncertainty in the calculated area will change, as the uncertainty in d is a factor in the calculation.

In summary, the use of constants in scientific formulas is necessary for accuracy and consistency, and the uncertainty in measurements of physical terms will be accounted for in the overall uncertainty of the calculated value.