Question

What would happen to the acceleration due to gravity, and to our acceleration toward the centre of the Earth, if the Earth stopped rotating?

Answer 1

If the earth stopped rotating, there would be no change in the acceleration due to gravity, and a negligible change in our acceleration toward the center of the earth.

Explanation:

There will be no change in the value of `g`, which depends only on the mass of the earth and our distance from it.

`g =(GM)/r^2` where:

`G` is the gravitational constant, `6.67xx 10^-11` `m^3 kg^-1 s^-2`,
`M` is the mass of the earth `5.97xx10^24` `kg` and
`r` is our distance from the center of the earth when we're standing on the surface, `6.37xx10^6` `m`.

This calculation yields the familiar value `g=9.8` `ms^-2`.

There will be a very small change in the net downward force acting on an object to hold it to the earth because the centripetal acceleration toward the center of the earth caused by the earth's rotation would disappear if the earth was no longer rotating.

The centripetal acceleration is given by `a=v^2/r` in linear terms or `a= omega^2r` in rotational terms. The radius of the earth is as above, `6.37xx10^6` `m`, and the angular velocity is `2pi` radians every 24 hours, which is `2.3xx10^-5` `rad` `s^-1`.

Using these values in the equation:

`a= omega^2r` = `(2.3xx10^-5)^2*6.37xx10^6 = 3.4 xx 10^-3` `ms^-2`.

Compared with a value of `9.81` `ms^-2` for the acceleration due to gravity, this is on the order of 3,000 times smaller, so we can really neglect it.