How Does Distance from Earth's Center Affect Gravity?

[King Khan]

Homework Statement

The force of gravity at Earth's surface on an astronaut if 634N. What is the force of gravity on the same person if the distance is 2m, in multiplies of Earth's radius, from the center of Earth?

Homework Equations

F=Gm1m2/r^2

The Attempt at a Solution

I really don't understand what to plug in where seeing as how they don't even give the mass, if someone could just please help me with this part I could do the rest.

 

[Dale]

Why don't you write it as a ratio of two forces and see if anything cancels.

Fa/Fb

Where Fb is the force at the surface and Fa is the force at twice the radius.

 

[King Khan]

Why don't you write it as a ratio of two forces and see if anything cancels.

Fa/Fb

Where Fb is the force at the surface and Fa is the force at twice the radius.

if you could rephrase that, sorry but I really don't understand.

 

[Dale]

$$F_a=\frac{G \, m1_a \, m2_a}{r_a^2}$$
$$F_b=\frac{G \, m1_b \, m2_b}{r_b^2}$$
Write Fa/Fb, simplify and solve.

 

[rwisz]

To calculate the force of gravity on the astronaut at a distance of 2m from the center of the Earth, we first need to determine the mass of the astronaut. We can use the given information that the force of gravity at Earth's surface on the astronaut is 634N.

We can use the formula F = mg, where F is the force of gravity, m is the mass of the astronaut, and g is the acceleration due to gravity. At Earth's surface, g is approximately 9.8 m/s^2.

So, we can rearrange the formula to solve for m: m = F/g. Plugging in the given force of gravity (634N) and the value of g (9.8 m/s^2), we get a mass of approximately 64.69 kg.

Now, we can use the formula F = Gm1m2/r^2 to calculate the force of gravity at a distance of 2m from the center of the Earth. We already know the mass of the astronaut (m2) and the distance from the center of the Earth (r=2m). We also know that m1 is the mass of the Earth, which is approximately 5.972 x 10^24 kg.

Plugging in all the values, we get:
F = (6.67 x 10^-11 Nm^2/kg^2)(5.972 x 10^24 kg)(64.69 kg) / (2m)^2
= 6.76 x 10^-6 N

Therefore, the force of gravity on the astronaut at a distance of 2m from the center of the Earth is approximately 6.76 x 10^-6 N. This is significantly less than the force of gravity at Earth's surface, which makes sense as the force of gravity decreases as we move further away from the center of the Earth.