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R^2 = (G m) / g
where radius equals the square root of (G times mass) divided by g
m = (gR^2) / G
where mass equals (g times radius squared) divided by G
and
g = (G m) / R^2
where g is (G times mass) divided by R squared
as for the gravitational constant
G = 6.67 x 10^(-11)
G is 6.67 times 10 to the -11 power
R^2 = (G m) / g
where radius equals the square root of (G times mass) divided by g
m = (gR^2) / G
where mass equals (g times radius squared) divided by G
and
g = (G m) / R^2
where g is (G times mass) divided by R squared
as for the gravitational constant
G = 6.67 x 10^(-11)
G is 6.67 times 10 to the -11 power
The expression for the mass of the planet in terms of the radius of the planet, acceleration due to gravity and gravitational constant is [tex]\boxed{{M_P} = \dfrac{{{(g_p)}{r^2}}}{G}}[/tex].
Further explanation:
Here, we have to derive the expression for the mass of the planet in terms of the radius of the planet, acceleration due to gravity and gravitational constant.
From the Newton’s law of the gravitation, gravitational force exerted by the planet on the object at the surface of the planet is directly proportional to the product of the mass of the planet and the object and inversely proportional to the square of the distance between them.
It can be written mathematically as,
[tex]\boxed{F = \frac{{G{M_e}{m_o}}}{{{r^2}}}}[/tex]
Here, [tex]G[/tex] is the gravitational constant and its value is [tex]6.674 \times {10^{ - 11}}\text{ m}^3/\text{kg}\cdot\text{s}^2[/tex], [tex]{M_e}[/tex] is the mass of the planet, [tex]{m_o}[/tex] is the mass of object, [tex]r[/tex] is the radius of the planet.
So, the gravitational force exerted by the planet on the object of unit mass at the surface of the planet can be calculated as,
[tex]{F_1}=\dfrac{{G{M_P}}}{{{r^2}}}[/tex]
As we know, the acceleration due to gravity is equal to the gravitational force or weight of the unit mass of object.
So, the acceleration due to gravity of the planet can be expressed as,
[tex]{g_p} = {F_1}[/tex]
Substitute the value of the [tex]{F_1}[/tex] as [tex]\dfrac{{G{M_P}}}{{{r^2}}}[/tex] in the above equation.
[tex]{g_p}=\dfrac{{G{M_P}}}{{{r^2}}}[/tex]
Simplify the above equation for mass of the planet,
[tex]\boxed{{M_P} = \frac{{{g_p}{r^2}}}{G}}[/tex]
Thus, the expression for the mass of the planet in terms of the radius of the planet, acceleration due to gravity and gravitational constant is [tex]\boxed{{M_P} = \dfrac{{{(g_p)}{r^2}}}{G}}[/tex].
Learn more:
1. Acceleration of object against friction: https://brainly.com/question/7031524
2. Change in the behavior of object due to change in gravity: https://brainly.com/question/10934170
Answer detail:
Grade: Senior School
Subject: Physics
Chapter: Gravitation
Keywords:
Derive a formula, mass of planet, radius, gravity, gravitational constant , gravitation, Earth, 6.67x10^-11, acceleration due to gravity, force of attraction, object.
