# Minimum Earth’s angular velocity to which we’d have no weight on a scale

Consider the Earth’s axis of rotation (on itself) as the “z-axis” on a Cartesian coordinate system in which the center of the Earth corresponds to the origin of such a coordinate system. What is the minimum angular velocity or speed of the Earth to which we’d start showing 0 weight on a scale?

The answer depends on the latitude of the considered person. For example the required speed is lesser at the equator than any other place, and is infinite at both poles.

Neglecting the Earth’s motion around the Sun, considering the Earth’s surface as a perfectly spherical and assuming that the mass density inside the Earth has at most a radial dependency (so no dependency on neither the azimuthal nor zenithal angles), any “point” or let’s say “body” (except the 2 poles) on the surface are in circular motion. In such a motion the body is under acceleration, the centripetal acceleration whose magnitude is worth $\omega ^2 r$ where $\omega$ is the angular speed. This acceleration is equal to the gravity’s acceleration $g$ in the case of the critical angular speed of rotation to which the body would start showing no weight on a scale. In such a case we have the relation $\omega _{\text{critical}}=\sqrt{\frac{g}{r}}$ where $r$ is the distance of the body to the axis of rotation. In spherical coordinates, $r=R \sin \theta$ where R stands for the Earth’s radius and theta is the zenithal angle.

Taking $g=9.8 \text{ m/s}^2$ and $R=6367444.7 \text{ m}$ and considering the equator latitude (theta equal to pi/2), we get $\omega_{\text{critical}}\approx 1.24 \times 10 ^{-3} \text{ rad/s}$. While the current Earth’s angular speed is roughly worth $\frac{2\pi \text{ rad}}{23 \text{ hours } 56 \text{ min}} \approx 7.27 \times 10^{-5} \text{ rad/s}$.

From which we get a ratio $\frac{\omega_{\text{critical}}}{\omega_{\text{Earth}}} \approx 17$. In other words the Earth would have to rotate on itself about 17 times faster than it currently does in order for people to show no weight on a scale at the equator.