Forces on us on a rotating Earth

Question

So the earth is constantly rotating but it doesn't need a force to rotate. It'll rotate indefinitely.(?) But we and other masses on earth need a force on us to continue rotating along with earth? And if earth rotated a lot faster then we would notice it because we would not be planted to same place on earth?

Answer 1

The Earth doesn't need a force to make it continue to rotate- quite the opposite: it will continue to rotate unless a force is applied to stop it.

In fact the Earth's rotation about its axis will eventually slow to a halt as a consequence of its interaction with the Moon, but that will take billions of years.

The force that keeps us planted to the ground and moving with the Earth is gravity. If the Earth spun much faster, then not only would we float off, but the Earth itself would break up as gravity would no longer be sufficient to provide the centripetal force to hold its constituent parts together.

Answer 2

Assuming the earth has no external torque applied to it then the earth would rotate indefinitely. This is the conservation of angular momentum at play.

We stay on the earth's surface because we also have angular momentum, angular momentum such that our angular velocity matches that of the earth. If we didn't we would feel a drag force due to the mismatch of our velocity and the surface of the earth's velocity.

Gravity is the radial force that keeps us in this circular motion co-moving with the earth's surface. In fact, the gravity we experience is more than required to keep us on this trajectory.

Consider the forces acting on a person standing on the equator of the earth. There is a balance between the gravitational force and the normal reaction force (the ground pushing us back up due to electrostatic repulsion).

$$m \ddot{r}=N-mg$$

For the person to stay rotating around the earth with the earths angular velocity $\omega$, we have the centripetal motion equation

$$m\ddot{r} = -m\omega^2r$$

Substituting this into our first expression we find

$$N=mg - m\omega^2r$$

This is important since when we say we feel gravity, we are really talking about the normal reaction force, which is the force we feel on our body when we stand still. In the case of the earths rotation, this effect is small, $N \approx mg$.

If the earth were to rotate faster, we see the value of $N$ would decrease. As $\omega^2 \rightarrow \frac{g}{r}$ the observer c0-moving with the earth's surface would experience a smaller and smaller 'effective gravity' (which is just a consequence of being in a non-inertial frame) until experiencing almost weightlessness.

Aside: It's worth noting, one; that is effect depends on your latitude. As we move towards the north/south pole, the distance from the observer to the axis of rotation, $r$, gets smaller and smaller, until at the poles $r=0$ and so $N=mg$. Again for our case, this effect is small but if the earth were the rotate much faster then this would be important.

Second; the earth is a large dynamical mass of rock that isn't perfectly spherical, thus the moon's gravity can pull on different parts of the earth with slightly different strengths (due to the earth's mass distribution) + the tides form by the moon's orbit can generate large frictional forces on the earth. This results in a net torque on the earth which does slow down the earth's rotation. In fact, the earth's orbital time period 1 billion years ago was roughly only 18 hours. We approximate it as rotating at a constant angular velocity since the time scales involved with problems we care about are much less than the time required for the earth's angular velocity to significantly slow down.