CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 6, Exercise 1 Solution
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Comparing Newton's Law of Gravitation to W = mg

PROBLEM
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Calculate the gravitational force between the Earth (mass 5.97 x
10^24 kg) and a 70 kg person standing on its surface (radius 6.371 x
10^6 m), using F = Gm1m2/r^2. Compare your answer to the person's
weight calculated using W = mg from Chapter 2 (use g = 9.81 m/s^2).
Are the two values close?

SOLUTION
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Step 1: Calculate the gravitational force using Newton's law of
universal gravitation.

    F = G m1 m2 / r^2

    G = 6.674 x 10^-11 m^3 kg^-1 s^-2
    m1 = 5.97 x 10^24 kg (Earth)
    m2 = 70 kg (person)
    r = 6.371 x 10^6 m

    F = (6.674 x 10^-11) x (5.97 x 10^24) x 70 / (6.371 x 10^6)^2
    F = (6.674 x 10^-11) x (4.179 x 10^26) / (4.059 x 10^13)
    F = 2.79 x 10^16 / 4.059 x 10^13
    F = 687.1 N (approximately)

Step 2: Calculate weight using W = mg.

    W = mg
    W = 70 x 9.81
    W = 686.7 N

ANSWER: F (from the full gravitation formula) is approximately
687.1 N, and W = mg gives approximately 686.7 N - the two values
are extremely close (differing only by rounding).

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WHY THIS WORKS AS AN ANSWER

W = mg is not a separate, competing formula from Newton's law of
universal gravitation - it is a simplified, practical shortcut for
exactly this everyday situation: an object sitting on a much larger
body's surface. The "g" in W = mg is really just GM/r^2 for that
specific body (Earth, in this case) already worked out as a single
number, since M and r barely change for anything standing on Earth's
own surface. This exercise shows the two approaches genuinely agree
once M, G, and r are substituted in explicitly - W = mg is Newton's
full law, quietly pre-simplified for the one specific, extremely
common case of standing on a planet's surface.