ω1=(γM)/(R^2)=(6.67 ×10^-11 ×5 ·96 ×10^24)/((6.37 ×10^6)^2)=9.8 m / s^2 ω2=ω^2 R=((2 π)/(T))^2

step 1 Here I can write W1 is equal to G M .E. by R .E. square. W1 is acceleration due to gravity. M .E

Ω1=(γM)/(R^2)=(6.67 ×10^-11 ×5 ·96 ×10^24)/((6.37...

$\omega_{1}=\frac{\gamma M}{R^{2}}=\frac{6.67 \times 10^{-11} \times 5 \cdot 96 \times 10^{24}}{\left(6.37 \times 10^{6}\right)^{2}}=9.8 \mathrm{~m} / \mathrm{s}^{2}$ $\omega_{2}=\omega^{2} R=\left(\frac{2 \pi}{T}\right)^{2} R=\left(\frac{2 \times 22}{24 \times 3600 \times 7}\right)^{2} 6.37 \times 10^{6}=0.034 \mathrm{~m} / \mathrm{s}^{2}$ and $\omega_{3}=\frac{\gamma M_{S}}{R_{\text {mean }}^{2}}=\frac{6.67 \times 10^{-11} \times 1.97 \times 10^{30}}{\left(149.50 \times 10^{6} \times 10^{3}\right)^{2}}=5.9 \times 10^{-3} \mathrm{~m} / \mathrm{s}^{2}$ Then $\quad \omega_{1}: \omega_{2}: \omega_{3}=1: 0.0034: 0.0006$

$\omega_{1}=\frac{\gamma M}{R^{2}}=\frac{6.67 \times 10^{-11} \times 5 \cdot 96 \times 10^{24}}{\left(6.37 \times 10^{6}\right)^{2}}=9.8 \mathrm{~m} / \mathrm{s}^{2}$
$\omega_{2}=\omega^{2} R=\left(\frac{2 \pi}{T}\right)^{2} R=\left(\frac{2 \times 22}{24 \times 3600 \times 7}\right)^{2} 6.37 \times 10^{6}=0.034 \mathrm{~m} / \mathrm{s}^{2}$
and $\omega_{3}=\frac{\gamma M_{S}}{R_{\text {mean }}^{2}}=\frac{6.67 \times 10^{-11} \times 1.97 \times 10^{30}}{\left(149.50 \times 10^{6} \times 10^{3}\right)^{2}}=5.9 \times 10^{-3} \mathrm{~m} / \mathrm{s}^{2}$
Then $\quad \omega_{1}: \omega_{2}: \omega_{3}=1: 0.0034: 0.0006$

Key Concepts

Angular Velocity and Period Relationship

Angular velocity is a measure of the rate of rotation, defined as the angle rotated per unit time, and is inversely related to the period of rotation. The relationship is given by ? = 2?/T, where T is the period. This concept allows for converting between how fast an object rotates (or orbits) and the time it takes to complete one full rotation, an essential step when analyzing rotational dynamics in celestial mechanics.

Centripetal Acceleration

Centripetal acceleration refers to the acceleration that is required to keep an object moving in a circular path and is directed towards the center of the circle. It is calculated as a = ?²R, where ? is the angular velocity and R is the radius of the circular path. This concept is important in analyzing the effects of rotation, such as the Earth’s rotation around its axis or a satellite’s motion around a planet.

Universal Law of Gravitation

This law expresses that every pair of masses in the universe attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It is mathematically described by F = G·(M1·M2)/r², where G is the gravitational constant. This concept underpins calculations of gravitational acceleration at the surface of a body as well as the gravitational influence of another body at a given distance.

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