The center of a moon of mass m is a distance d from the center of a planet of mass m. at some distance x from the center of the planet, along a line connecting the centers of planet and moon, the net force on an object will be zero. a) derive an expression for x in terms of m, m, and d. b) if the net force is zero a distance ⅔d from the planet, what is the ratio r of the mass of the planet to the mass of the moon, m/m?

A) Th expression for x in terms of m, M and D is;

x =

B) The ratio R of the mass of the planet to the mass of the moon is;

R = M/m = 4

We are given;

Mass of moon = m

Mass of planet = M

Center to center distance from moon to planet = D

Net force is zero at a distance x from the planet's center.

F_g = GMm/r²

Now, since x is a distance from the center of planet, then its' distance from

the moon is; D - x

This means;

For the moon; F_g,moon = Gm₁m/(D - x)²

For the planet; F_g,planet = Gm₁M/x²

Where m₁ is mass of the earth

We are told that net force on the object is zero.

Thus; F_g,moon = F_g,planet

⇒ Gm₁m/(D - x)² = Gm₁M/x²

G and m₁ will cancel out to give us;

m/(D - x)² = M/x²

Cross multiply to get;

mx² = M(D - x)²

Taking square root of both sides gives;

(√m)x = (√M) × (D - x)

Expand the right side to get;

(√m)x = D(√M) - x(√M)

(√m)x + x(√M) = D(√M)

x((√m) + (√M)) = D(√M)

x =

B) Net force is at ⅔D from the planet. Thus, x = ⅔D from the planet.

From earlier, we saw that; mx² = M(D - x)². Thus;

M/m = x²/(D - x)²

Putting ⅔D for x gives;

M/m = (⅔D)²/(D - ⅔D)²

M/m = ⁴/₉(D²)/(¹/₉D²)

This will simplify out to give;

M/m = 4

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Answer with Explanation:

Let  rest mass at point P  at  distance x from center of the planet, along a line connecting the centers of planet and the moon.

Mass of moon=m

Distance between the center of moon and center of planet=D

Mass of planet=M

We are given that net force on an object will be zero

a.We have to derive an expression for x in terms of m, M and D.

We know that gravitational force=

Distance of P from moon=D-x

=Force applied on rest mass due to m

=Force on rest mass due to mas M

because net force is equal to 0.

Let

Then,

b.We have to find the ratio R of the mass of the mass of the planet to the mass of the moon when x=

Net force is zero

Hence, the ratio R of the mass of the planet to the mass of the moon=4:1

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