Question

Planet A has mass 3M and radius R, while Planet B has mass 4M and radius 2R. They are separated by center-to-center distance 8R. A rock is released halfway between the planets’ centers at point O. It is released from rest. Ignore any motion of the planets
. - Calculate the magnitude of the rock’s acceleration, in meters per second squared, for M = 7.3×1023 kg and R = 5.8×106 m.

Answer 1

Answer:

Explanation:



In Newton's law of universal gravitation

F = Gm₁m₂/r²

Where G is a gravitational constant = 6.674e-11m³/kgs²

m₁ and m₂ are the masses of the two bodies or objects in question, in kilogram (kg)

r is the distance in meters between them

From the question, the rock is placed halfway between the planets

So, it's distance from planet A is 8R/2 = 4R

And it's distance from planet B is also 8R/2 = 4R

Using F = Gm₁m₂/r²

To Planet A

r = 4R,

m₁ = mass of Rock = m

m₂ = mass of planet A = 3M

So Fa = G mm₂/r² = Gm(3M) / (4R)²

To Planet B,

r = 4R,

m₁ = mass of Rock = m

m₂ = mass of planet B = 4M

Fb = G mm₂/r² = Gm(4M) / (4R)²

Comparing both forces together, we realise that Planet B has the largest force,

so take we F = Fb – Fa

F = Fb – Fa = Gm(4M) / (4R)² – Gm(3M) / (4R)²

F = GmM/16R²)(4–3)

F = GmM/16R²

Note that Force = Mass * Acceleration

So, F = ma

So, ma = GmM/16R² ------- Divide through by m

a = GM/16R²

From the question

M = 7.3×10^23kg

R = 5.8×10^6 m

So, a = (6.674 * 10^-11 * 7.3×10^23)/16(5.8×10^6)²

a = (48.7202 * 10^12)/16(33.64 * 10^12)

a = (48.7202 * 10^12)/(538.4 * 10^12)

a = 48.7202/538.4

a = 0.090517612960760

a = 0.091m/s² ----------Approximated