Circumference of Circle with Uncertainty

In summary, the conversation discusses calculating the circumference of a circle with a measured radius of 7.3 ± 0.2 cm. Two methods are presented, with the first one being the recommended approach using calculus. The final answer is determined to be 46 ± 1 cm, with the error being calculated using the formula ΔC = 2πr Δr. It is advised to not round down errors and to include an additional digit when the first non-zero digit is 1 or 2. The number of significant digits becomes less relevant when the error is included in the answer.
  • #1
ELLE_AW
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Homework Statement


Calculate the circumference (including uncertainty) of a circle whose measured radius is r=7.3 ± 0.2cm.

2.Relevant equations & 3.The attempt at a solution

- Circumference of circle --> C = 2πr = 2π7.3 = 45.87 cm

- Exact constant error propagation --> z = kx

- Limit Error --> δz = kδx

- Therefore, δC = 2π(0.2 cm) = 1.257 cm

Final Answer = 46 ± 1 cmOR Should I be going about it like this:

- Circumference of circle --> C = 2πr = 2π7.5 = 47.124 cm = 47 cm

- Circumference of a circle --> C = 2πr = 2π7.1 = 44.611 cm = 45 cm

- Final Answer 46 ± 1 cmI was discussing this with someone else. I did it the first way and they did it the second way.

Also, my final answer has the correct significant figures right?
 
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  • #2
You should do it the first way. Bracketing the calculated value with the minimum and maximum values using the given uncertainty is for people who are not conversant with calculus and do not understand where ΔC = 2πr Δr comes from. Yes, your final answer has the correct sig. figs.
 
  • #3
kuruman said:
You should do it the first way. Bracketing the calculated value with the minimum and maximum values using the given uncertainty is for people who are not conversant with calculus and do not understand where ΔC = 2πr Δr comes from. Yes, your final answer has the correct sig. figs.
Thank you! That's what I thought as well :)
 
  • #4
Word of warning: You typically should not round errors down. This overrepresents the accuracy that you have. The typical thing to do when you have an error whose first non-zero digit is 1 or 2 is to include another digit to also avoid overstating the error.

The number of significant digits is not very relevant when you include the errors, the entire point of significant digits is that it is kind of a poor man's error analysis - letting you get a feeling for the kind of accuracy that you have from the number of digits you have included - but this is obsolete when you have the actual error!

In this case, I would answer 45.9±1.3 cm.
 

Related to Circumference of Circle with Uncertainty

1. What is the formula for calculating the circumference of a circle with uncertainty?

The formula for calculating the circumference of a circle with uncertainty is C = 2πr ± u, where C is the circumference, π is the mathematical constant pi, r is the radius of the circle, and u is the uncertainty or margin of error.

2. What is the significance of calculating the circumference of a circle with uncertainty?

Calculating the circumference of a circle with uncertainty allows us to determine the range of possible values for the circumference, taking into account any potential errors or variations in the measurement of the radius. This gives a more accurate representation of the true value of the circumference.

3. How is uncertainty in the circumference of a circle typically expressed?

Uncertainty in the circumference of a circle is typically expressed as a plus or minus value, indicating the range of possible values for the circumference. For example, a circumference of 10 cm ± 0.5 cm means that the true value of the circumference could be between 9.5 cm and 10.5 cm.

4. How does the uncertainty in the radius affect the calculated circumference of a circle?

The uncertainty in the radius of a circle directly affects the uncertainty in the calculated circumference. A larger uncertainty in the radius will result in a larger uncertainty in the circumference, and vice versa.

5. Can the uncertainty in the circumference of a circle be reduced?

Yes, the uncertainty in the circumference of a circle can be reduced by improving the precision of the measurements used to calculate the radius. This can be achieved through more accurate measuring tools or by taking multiple measurements and calculating an average.

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