In an experiment to determine the density of a solid sphere, the mass of the sphere is mea… — JAMB UTME

In an experiment to determine the density of a solid sphere, the mass of the sphere is measured as $(50.0 \pm 0.5)\text{ g}$ and its radius is measured as $(1.00 \pm 0.02)\text{ cm}$ . What is the maximum percentage error in the calculated density of the sphere?

Cevap: 7 %

Cevap

The maximum percentage error in the calculated density of the sphere is

7.0\%

Density is related to mass and radius by

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

. In error analysis, the maximum fractional error of a calculated quantity is the sum of the fractional errors of its components multiplied by their respective powers. The mass has a percentage error of

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

, and the radius has a percentage error of

\frac{0.02}{1.00} \times 100\% = 2.0\%

. Multiplying the radius percentage error by 3 gives

6.0\%

, and adding the mass percentage error of

1.0\%

yields a maximum percentage error of

7.0\%

Adım Adım Çözüm

Determine the percentage error in the measurement of mass (

m

Percentage error in mass =

\frac{0.5\text{ g}}{50.0\text{ g}} \times 100\% = 1.0\%

Relative error multiplied by 100 gives the percentage error of a measurement.

Determine the percentage error in the measurement of radius (

r

Percentage error in radius =

\frac{0.02\text{ cm}}{1.00\text{ cm}} \times 100\% = 2.0\%

Relative error in radius multiplied by 100 gives its percentage error.

Apply the error propagation formula for the density of a sphere.

Maximum percentage error in density =

1.0\% + 3(2.0\%) = 7.0\%

Density is given by

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

. For a formula of the form

X = A^a B^b

, the fractional error propagates as

\frac{\Delta X}{X} = a\frac{\Delta A}{A} + b\frac{\Delta B}{B}

. Here, the exponent of

r

is 3, so its percentage error is multiplied by 3.

Anahtar Kavram

Error propagation in fractional powers and derived physical quantities

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