Relation to g = 9.81 m/s²

The law of universal gravitation does not replace the formula $F_\text{peso} = mg$ we already know: it contains it. Near the Earth’s surface, the force with which the Earth attracts a body of mass $m$ is precisely its weight, and from this we obtain the deeper meaning of $g$: it is not a mysterious constant, but the gravitational field produced by the Earth at its surface.

Writing the universal law for a body resting on the Earth (mass $M_T$, radius $R_T$):

$F = G\frac{M_T\,m}{R_T^2} = m\underbrace{\frac{GM_T}{R_T^2}}_{g}$

The factor multiplying $m$ is exactly $g$. Substituting $M_T = 6\cdot 10^{24}\;\text{kg}$ and $R_T = 6{,}4\cdot 10^6\;\text{m}$:

$g = \frac{6{,}67\cdot 10^{-11}\cdot 6\cdot 10^{24}}{(6{,}4\cdot 10^6)^2} = 9{,}78\;\text{m/s}^2$

The experimentally measured value is $9{,}81\;\text{m/s}^2$; the small difference is due to the Earth’s rotation and its not-quite-spherical shape.

Key formula

$\ev{g = \frac{GM_T}{R_T^2}}$ The acceleration due to gravity at the surface depends on the planet’s $M_T$ and $R_T$: on another celestial body it takes a different value.

This relation has a remarkable practical consequence: weighing a body at a planet’s surface and measuring its radius lets us work out the planet’s own mass. This is the gateway to gravitation as a tool for cosmic measurement.

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Topics: Gravitazione Concepts: Legge di gravitazione universale · Forza peso Skills: Impostazione simbolica

Related exercises: The hollow Moon · Force between two 100 kg spheres · Gravity on the Moon and an astronaut’s weight