Question:

The gravitational force between the moon and Earth creates a tidal force. From the figure, \(a\) is the distance between the moon and the Earth. \(M\) and \(m\) are the masses of Earth and moon, respectively. \(r\) denotes the radius of Earth. Find the differential tidal acceleration.

Answer:

The tidal force is obtained by the difference of gravitational fields between C (center of mass) and S (place to get tidal force). This can be associated with the differential tidal acceleration. Let us write down each gravitational acceleration.

\[  g_C = \frac{Gm}{a^2}  \]

\[  g_S = \frac{Gm}{(a+r)^2}  \]

The difference of them is the tidal acceleration.

\[  g_C -g_S = \frac{Gm}{a^2} – \frac{Gm}{(a+r)^2}  \]

\[   = \frac{Gm}{a^2}\left(1-\frac{a^2}{(a+r)^2}\right)  \]

\[  = \frac{Gm}{a^2}\left(1-\frac{1}{(1+\frac{r}{a})^2}\right)  \]

\[   \sim \frac{Gm}{a^2}\left(1-\left\{1-2\frac{r}{a}\right\}\right)  \]

The above uses approximation. Hence, we have

\[  g_{\mathrm{tidal}} = \frac{2Gmr}{a^3}  \]