The gravitational potential energy of two point masses separated by is

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Calculus of change and accumulation

Mathematical Methods for Physics, Ch The derivative as a physical instrument

Problem

The gravitational potential energy of two point masses separated by rr is U(r)=−GMmrU(r) = -\dfrac{GMm}{r}, with UU in joules and rr in metres. Someone works out the force between them as follows: "dUdr=−GMmr\dfrac{\mathrm{d}U}{\mathrm{d}r} = -\dfrac{GMm}{r}, so the force is F=−dUdr=GMmrF = -\dfrac{\mathrm{d}U}{\mathrm{d}r} = \dfrac{GMm}{r}." ((a)) Check the units of the claimed derivative dU/dr\mathrm{d}U/\mathrm{d}r against the units a derivative of an energy with respect to a length must have. ((b)) State the calculus error that produced the wrong result, and give the correct expression for dU/dr\mathrm{d}U/\mathrm{d}r and for FF.

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