For small heights above Earth’s surface

PHYSICS · Multi-part · CBSE · Physics · Units and Measurements · Units and Measurements

Problem

For small heights

h \ll R_E

above Earth’s surface, two classmates propose different approximate formulas for the change in gravitational potential energy

\Delta U

of an object of mass

m

raised through a height

h

: \newline ewlineFormula A:

\Delta U = \dfrac{G M_E m h}{R_E^2}

ewlineFormula B:

\Delta U = m g h

(a) Using dimensional considerations only, identify which of the two formulas (A or B) has the correct dimensions of energy. (b) Take

M_E = 6 \times 10^{24}\,\text{kg}

R_E = 6.4 \times 10^{6}\,\text{m}

g = 9.8\,\text{m/s}^2

m = 2\,\text{kg}

and

h = 5\,\text{m}

. Calculate

\Delta U

using both formulas. (c) Compare your numerical answers and comment (in not more than two lines) on the suitability of the approximate formula you identified in part (a) for small heights.

Related theory

Gravitational Potential Energy
Gravitation
Dimensional Analysis
Deriving relations using dimensions
Dimensional analysis and its applications

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