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.