# RMS error in volume of a cylinder

• OONeo01
In summary, the length and radius of a perfect cylinder are measured with an RMS error of 1%. The RMS error of the inferred volume of the cylinder is 3%.
OONeo01

## Homework Statement

The length and radius of a perfect cylinder are measured with an RMS error of 1%. The RMS error of the inferred volume of the cylinder is.. ?

## Homework Equations

V=∏r2h
Hence dV/V=2dr/r+dh/h

## The Attempt at a Solution

I took dr/r and dh/h as 1%. So got the final answer as 2x1%+1%=3%
But the question came with 4 options(one of which is correct):
1.7%, 3.3%, 0.5% and 1%

So my answer is wrong ! What am I overlooking ? Is there any special precautions to be taken about RMS error ?

The uncertainties of the two measurements should be independent, so I would add those contributions in quadrature and not linear. But then the result is 2.2% :D.
1.7% would be the answer if all three dimensions had been measured independently.

mfb said:
1.7% would be the answer if all three dimensions had been measured independently.

Could you please explain that, or give some link where it has been explained ? I didn't quiet understand what you meant by that.. X-)

The two errors might add up, but also they might partly cancel. So the RMS error in the volume will be less than simply 3 x one dimension of error. Instead, you use a root-sum-squares way of adding them up. But mfb is right that the answer should not be 1.7 either. That would be the answer for volume of a rectangular block with independent 1% errors in each of the three dimensions: √(12+12+12) = √3.
In the present case, there are only two measurements. An error of x% in the radius will produce an error of 2x% in the cross-sectional area, so we have √(12+22) = √5 ≈ 2.2, which is not in the list.

Ok. So √3 which matches with one of the given options doesn't really make much sense.

Thanks a lot for the help guys. What was more important to me was how to find the RMS error(of any measurement, not just the volume of a cylinder) rather than just matching answers with options. I guess I learned that. :-)

## 1. What is RMS error in the volume of a cylinder?

RMS error in the volume of a cylinder is a metric used to measure the accuracy of a calculated volume compared to the true volume of the cylinder. It takes into account both the magnitude and direction of the errors.

## 2. How is RMS error in volume of a cylinder calculated?

RMS error in volume of a cylinder is calculated by taking the square root of the sum of the squared errors, divided by the number of measurements. This is also known as the root mean square deviation (RMSD).

## 3. What does a high RMS error in volume of a cylinder indicate?

A high RMS error in volume of a cylinder indicates that there is a large amount of variation between the calculated volume and the true volume of the cylinder. This could be due to imprecise measurements or inaccurate calculations.

## 4. How can RMS error in volume of a cylinder be minimized?

RMS error in volume of a cylinder can be minimized by ensuring accurate and precise measurements of the cylinder's height and radius, as well as using the correct formulas for calculating volume. Double-checking calculations and using more precise measuring tools can also help reduce RMS error.

## 5. Is RMS error in volume of a cylinder the only measure of accuracy?

No, RMS error in volume of a cylinder is not the only measure of accuracy. Other metrics, such as mean absolute error and mean squared error, can also be used to evaluate the accuracy of calculated volumes. It is important to consider multiple measures of accuracy for a more comprehensive understanding.

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