Question

How many times greater is the force of gravity on a 4 kg object lying on the surface of a moon than on a 4 kg object orbiting at a distance of 9 moon radii above the surface?

Answer 1

The force of gravity on a 4 kg object on the moon's surface is 81 times greater than on the same object orbiting at 9 moon radii above the surface, due to the inverse square law of gravitational force.

Step-by-step explanation:

The student is asking about the comparative force of gravity acting on a 4 kg object that is on the surface of the moon versus one orbiting at a distance of 9 moon radii above the surface. To find how many times greater the force of gravity is on the object on the moon's surface, we use Newton's law of universal gravitation, which states that the force of gravity is inversely proportional to the square of the distance between two masses. The acceleration due to gravity on the moon's surface is 1.67 m/s². When an object is 9 moon radii away, the force of gravity is reduced by a factor of (1/9)².

Let's denote the gravitational force at the moon's surface as F₁ and the force at 9 moon radii above as F₂. Since the gravitational force is proportional to 1/r², we have:

• F₁ is proportional to 1/R² (where R is the moon's radius)
• F₂ is proportional to 1/(9R)²

Now, the ratio of F₁ to F₂ is:

F₁ / F₂ = (1/R²) / (1/(9R)²) = 9² = 81

Therefore, the force of gravity on a 4 kg object on the moon's surface is 81 times greater than on a 4 kg object orbiting at 9 moon radii above the surface.

Answer 2

Force is 81 times greater.

Step-by-step explanation:

Let us start by stating the formula given by Newton, the Universal Law of Gravity:

`F_(g)=G(Mm)/(r^2)`,

where Fg is the force of gravity applied between two bodies of masses M (central body) and m (object). G is th universal constant of gravity, and r stands for the distance between M and m bodies' centers of gravity.

Since m is the same value for both cases, Moon's mass M remains the same (constant) and G is a constant value, the relation between the two forces asked in the question can be stated as:

`(F_(g1))/(F_(g2))=((r_(2))^2)/((r_(1))^2)=((9R_(moon))^2)/((R_(moon))^2)=9^2=81`

Distance values r are here expressed as a proportion of moon radii. Bare in mind that an object on a planet's surface is one planet's radius away from the planet's center of gravity, so r=R.