Sphere Volume Error Calculation

Understanding how errors propagate in measurements is crucial in physics and engineering. This problem asks us to determine the percentage error in the volume of a sphere when there is a known percentage error in the measurement of its radius.

Volume of a Sphere Formula

The formula for the volume (\(V\)) of a sphere with radius (\(r\)) is given by:

\(V = \frac{4}{3}\pi r^3\)

Here, \(\frac{4}{3}\) and \(\pi\) are constants, meaning they have no error in their values for this calculation. The only quantity subject to measurement error is the radius \(r\).

Error Propagation Principle

When a physical quantity \(Y\) depends on another measured quantity \(X\) raised to a power, such as \(Y = kX^n\) (where \(k\) is a constant and \(n\) is the power), the fractional error in \(Y\) is related to the fractional error in \(X\) by the following principle:

\(\frac{\Delta Y}{Y} = n \frac{\Delta X}{X}\)

Where \(\Delta Y\) is the absolute error in \(Y\), \(\Delta X\) is the absolute error in \(X\), \(\frac{\Delta Y}{Y}\) is the fractional error in \(Y\), and \(\frac{\Delta X}{X}\) is the fractional error in \(X\).

To convert fractional error to percentage error, we simply multiply by 100%:

\(\frac{\Delta Y}{Y} \times 100\% = n \left(\frac{\Delta X}{X} \times 100\%\right)\)

Applying Error Calculation to Sphere Volume

Let's apply this principle to the volume of the sphere:

1. Identify the quantities:
   • The quantity we are measuring the error in is the volume, \(V\).
   • The quantity with the given error is the radius, \(r\).
2. Relate \(V\) and \(r\):
   • From the formula \(V = \frac{4}{3}\pi r^3\), we can see that \(V\) depends on \(r^3\).
   • Here, \(k = \frac{4}{3}\pi\) (a constant) and \(n = 3\) (the power of \(r\)).
3. Formulate the fractional error relationship:
   • Using the error propagation principle, the fractional error in volume (\(\frac{\Delta V}{V}\)) is related to the fractional error in radius (\(\frac{\Delta r}{r}\)) as follows:
   • \(\frac{\Delta V}{V} = 3 \frac{\Delta r}{r}\)
4. Convert to percentage error:
   • To find the percentage error in the measurement of the volume, we multiply both sides by 100%:
   • \(\frac{\Delta V}{V} \times 100\% = 3 \left(\frac{\Delta r}{r} \times 100\%\right)\)
5. Substitute the given value:
   • The problem states that the error in the measurement of the radius of the sphere is 1%.
   • So, \(\frac{\Delta r}{r} \times 100\% = 1\%\)
   • Substitute this value into our equation:
   • \(\frac{\Delta V}{V} \times 100\% = 3 \times 1\%\)
   • \(\frac{\Delta V}{V} \times 100\% = 3\%\)

Therefore, the error in the measurement of its volume is 3%.