Final answer:
The Earth's angular velocity, defined as its rotational speed around its axis, is one revolution per day, or approximately 0.000072722052 rad/s. If Earth loses mass and its radius decreases due to a catastrophic event, the new angular velocity can be found using conservation of angular momentum.
Step-by-step explanation:
Understanding the Angular Velocity of Earth
The angular velocity of the Earth is the rate at which the Earth rotates around its axis. It is expressed in terms of rotations or radians per unit of time. The Earth makes one complete rotation on its axis in a period of one day, which is equivalent to an angular velocity of one revolution per day. To perform calculations in SI units, this angular velocity must be converted into radians per second.
Using the data provided, if a layer of Earth with a mass of 1.0 × 10²³ kg breaks off, and thus decreases Earth's radius by 0.2 × 10¶ m, the conservation of angular momentum allows us to predict the change in Earth's angular velocity. Angular momentum (ℓ) is given by the product of the moment of inertia (I) and the angular velocity (ω), which for a solid sphere is I = 0.4 M R², assuming uniform density.
Initially, the angular momentum (L) of the Earth can be calculated as:
L = 0.4 M R² ω
After the change:
L' = 0.4 (M - ΔM) (R - ΔR)² ω'
The angular momentum before and after the event must be equal (L = L'), so by equating the two and solving for ω', we can find the new angular velocity.