DistanceToTheMoon

The distance to the moon can be computed as follows.

We know that:

acceleration due to gravity at the Earth's surface is about $9.8ms^{-2}.$

this can be measured using a pendulum in your living room

the orbital period of the moon relative to the stars is about 27.3 days.
acceleration due to gravity decreases as the inverse square.
the radius of the Earth is 6366km.

Measured by Eratosthenes

OK, so acceleration in a circle is $v^2/r$ or $\omega^2r$ where r is the radius of the orbit, and $\omega$ is the rotational velocity. That's 27.3 days divided by $2\pi$ giving 1/375402 radians/second. Hence acceleration in the orbit is $(1/375402)^2r$ where r is the (unknown) distance from the centre of the Earth to the Moon.

But acceleration due to gravity is $9.8(r/R)^{-2}$ where R is the radius of the Earth, so equating these we get:

$(1/375402)^2r=9.8(R/r)^2$

$r^3=9.8(R^2)(375402)^2$

This gives an answer of 382517km, which is amazingly close to the figure quoted on WikiPedia of an average centre-centre distance of 384,403km.

Accurate to 0.1%.

Finally, Orbital Velocity is given by $v=\omega{r}$ , and that now works out as 382517/375402 km/s, or almost exactly 1.02km/s.

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Distance To The Moon

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