Question 15: Units , Dimensions and Error Analysis

Question

In an experiment to measure the density of the material of a sphere, an error of $2 \%$ occurred while measuring the radius of a sphere and an error of $3 \%$ occurred while measuring the mass of the sphere. What is the maximum percentage error in the measurement of density?

Accepted Answer

To find the maximum percentage error in density, follow these steps:
Recall the density formula for a sphere:
$$ho = \frac{m}{V} = \frac{m}{\frac{4}{3}\pi r^3} \;\Longrightarrow\; ho \propto \frac{m}{r^3}.$$
Use linearized error propagation for products and powers:
$$\frac{\Delta ho}{ho} \approx \frac{\Delta m}{m} + 3\,\frac{\Delta r}{r}.$$
Plug in the given percentage errors (taking magnitudes for the Ã¢ÂÂmaximumÃ¢ÂÂ error):
Ã¢ÂÂ Mass error: $\Delta m/m = 3\%$
Ã¢ÂÂ Radius error: $\Delta r/r = 2\%$
Thus
$$\frac{\Delta ho}{ho} \approx 3\% + 3	imes2\% = 3\% + 6\% = 9\%.$$
Answer: 9 % (Option D).

Answer

8

Answer

6

Answer

7

Answer

9

Units , Dimensions and Error Analysis

In an experiment to measure the density of the material of a sphere, an error of 2 \% occurred while measuring the radius of a sphere and an error of 3 \% occurred while measuring the mass of the sphere. What is the maximum percentage error in the measurement of density?

Choose the correct answer: