In an experiment to determine the density of a solid sphere, the mass is measured as (50.0… — JAMB UTME

In an experiment to determine the density of a solid sphere, the mass is measured as $(50.0 \pm 0.5)\text{ g}$ and the radius is measured as $(2.00 \pm 0.05)\text{ cm}$ . What is the percentage error in the calculated density of the sphere?

Answer: 8.5 %

Answer

The percentage error in the calculated density of the sphere is

8.5\%

The density formula

\rho = \frac{3m}{4\pi r^3}

dictates that maximum relative error is given by

\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3\left(\frac{\Delta r}{r}\right)

. Evaluating the percentage errors gives

1.0\%

for mass and

2.5\%

for radius. Summing

1.0\% + 3(2.5\%)

yields

8.5\%

Step-by-Step Solution

Determine the functional dependence of density on measured quantities.

Density

\rho = \frac{m}{V} = \frac{3m}{4\pi r^3}

The volume of a sphere of radius

r

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

Formulate the maximum fractional error relationship.

\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3\left(\frac{\Delta r}{r}\right).

When combining uncertainties, fractional errors add, and exponents act as multiplying factors.

Calculate the percentage error in the mass measurement.

\frac{0.5}{50.0} \times 100\% = 1.0\%.

Percentage error in mass is the absolute uncertainty divided by the measured value multiplied by

100\%

Calculate the percentage error in the radius measurement.

\frac{0.05}{2.00} \times 100\% = 2.5\%.

Percentage error in radius is the absolute uncertainty divided by the measured value multiplied by

100\%

Calculate total percentage error in density.

Percentage error = 1.0\% + 3(2.5\%) = 8.5\%.

The radius contributes three times its relative error because volume depends on

r^3

Key Concept

Error propagation in derived quantities involving powers

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