Three artificial satellites—Alpha, Beta

PHYSICS · Multi-part · CBSE · Physics · Gravitation · Gravitation

Problem

Three artificial satellites—Alpha, Beta, and Gamma—orbit the same planet Z in nearly circular paths. The measured orbital radii

R

and orbital periods

T

are listed below. Satellite Data • Alpha:

R = 2.0\times10^{7}\,\text{m}

T = 2.0\times10^{4}\,\text{s}

• Beta:

R = 3.0\times10^{7}\,\text{m}

T = 3.4\times10^{4}\,\text{s}

• Gamma:

R = 4.0\times10^{7}\,\text{m}

T = 5.0\times10^{4}\,\text{s}

Assume the satellites’ masses are negligible compared with the mass of planet Z. (a) Using the data, describe the mathematical relationship that appears to exist between the orbital period

T

and the orbital radius

R

for the three satellites. Support your description with a brief quantitative comparison that uses at least two of the data pairs. (b) Explain, in terms of the forces acting on a satellite in uniform circular motion, why the relationship you identified in part (a) is expected for any satellite orbiting planet Z. (c) Identify one experimental uncertainty or modeling assumption that could cause small deviations from the exact relationship you described in part (a), and explain its possible effect on the data.

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