Question

Derive an expression for the mass of the earth, given the orbital period of the
moon and the radius of its orbit.

Answer 1

mass of the earth is given as M = 4π²r³/T²G .

Let mass of moon is m , mass of earth is M, radius of orbit is r and time period is T.

as we know, the moon is only one satellite of the earth which revolves around the earth in its orbit.

so, centripetal force = gravitational force

⇒mv²/r = GMm/r²

⇒v² = GM/r

for circular motion, v = ωr = (2π/T)r

then, (2πr/T)² = GM/r

⇒4π²r²/T² = GM/r

⇒ M = 4π²r³/T²G

hence, mass of the earth is given as M = 4π²r³/T²G

read more similar questions : Which graph best represents the variation of radius r of a circular orbit of a satellite against its time ...

https://brainly.in/question/12490999

The surface density (mass/area) of a circular disc of radius a depends on the distance from the center as ρ(r) = A+Br, F...

https://brainly.in/question/4586470

Answer 2

$\huge\bold\purple{Answer:-}$

Let mass of moon is m , mass of earth is M, radius of orbit is r and time period is T.

as we know, the moon is only one satellite of the earth which revolves around the earth in its orbit.

so, centripetal force = gravitational force

⇒mv²/r = GMm/r²

⇒v² = GM/r

for circular motion, v = ωr = (2π/T)r

then, (2πr/T)² = GM/r

⇒4π²r²/T² = GM/r

⇒ M = 4π²r³/T²G

hence, mass of the earth is given as M = 4π²r³/T²G