How Do Significant Figures Affect Calculations in Physics Problems?

[zed101]

Hi! I'm confused, please I would love some help with this:

Homework Statement

if the inner radius of a ring is 3,56 cm and the outer radius is measured as 3,32 cm, compute the area of the ring

Homework Equations

When multiplying the number of significant figures stays the same, when adding or subtracting we keep same number of decimals.
Area of a circle: \pi r^2

The Attempt at a Solution

I'm basically subtracting $\pi r_1^{2}-\pi r_2^2$. When squaring the radii I get three significant figures and when subtrating I keep two decimals. The answer that way has two decimals for a total of three significant figures, my answer would be $\pi 1.65 cm^2$ (pi has infinite significant figures). However, if I rearrange the formula as $\pi (r_1+r_2)(r_1-r_2)$ for $r_1-r_2$ I get 0, 24 cm which only has two significant figures, this crops the total number of significant figures down to two when multiplied with $r_1+r_2$. What am I doing wrong?

 

[haruspex]

Errors can become quite large, in proportion, when taking differences of similarly sized numbers. Suppose each of the original values has an error of +/- 0.005. The range of possible values for the difference of their squares is 1.58 to 1.72, so it's not even two significant digits.
This affected your calculation when you took the difference of the squares. Squaring produced 4 digits for each value, of which one disappeared when you took the difference. But after squaring each you should, technically, only have kept 3 sig figures in each, so when the difference lost the high order digit you should only have had 2 sig figures left.

 

[I like Serena]

Welcome to PF, zed101!

You're not doing anything wrong.
The rule is only a rule of thumb.

In particular, it breaks down when subtracting quantities with a result close to zero.
This is what happens in your case.

 

[zed101]

Hi! thanks for the replies! :) I see, so the rule of thumb is not always true, teachers never said that. I suppose I should go with the first method before I hand in my paper. I'll add that note about the problem hehe.
Thanks! I think I can work safely now :)

 

[Nickie]

Hello! It seems like you have encountered a common issue with significant figures. In this case, the ambiguity arises because you are using two different methods to calculate the area of the ring.

In the first method, you are using the formula for the area of a circle, which involves squaring the radius. This means that the number of significant figures in the radius will also be the number of significant figures in the area. In this case, your answer of \pi 1.65 cm^2 is correct because both radii have three significant figures.

However, in the second method, you are using the formula \pi (r_1+r_2)(r_1-r_2) which does not involve squaring the radius. This means that the number of significant figures in the radius does not necessarily match the number of significant figures in the area. In this case, your answer of 0.24 cm has only two significant figures because the difference between the two radii is only given to two decimal places.

So which method is correct? Both methods are actually correct, but they are giving you different levels of precision. The first method gives you a more precise answer because it takes into account the full number of significant figures in the radii. The second method gives you a less precise answer because it only takes into account the number of significant figures in the difference between the radii.

In science, it is important to use the appropriate number of significant figures to reflect the precision of your measurements. In this case, since the radii were measured to two decimal places, it is more appropriate to use the second method to calculate the area of the ring. However, if you had more precise measurements for the radii, then the first method would be more appropriate.

I hope this helps to clarify the ambiguity with significant figures. Keep in mind that significant figures are a way to represent the precision of your measurements, and it is important to use them correctly in scientific calculations.