A force of 5.25 newtons acts on an object of unknown mass at a distance of 6.9..*.108 meters from the center of Earth. To increase the force to 2.5 times its original value, how far should the object be from the center of Earth?

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matt101 · Answer

The nice thing about this question is that you don't actually need to worry about numbers until the very end! It's just a matter of setting up the right equation. First of all, remember Newton's law of universal gravitation:

$$F={Gm_1m_2 \over r^2}$$
Where F is the force of gravity between the two objects, G is the graviational constant, m(1) is the mass of one object, m(2) is the mass of the other object, and r is the distance between the two objects. We can rearrange this equation to solve for G instead (you'll see why this is important soon):

$$G={Fr^2 \over m_1m_2}$$
We have two situations here: (a) (the starting position of the objects) and (b) (the final position of the objects). We can use Newton's Law of gravitation to describe the force of gravity between these two objects during situations (a) and (b), but we can actually set these equations equal to one another because the value of G will be the same in both (it's a constant!) - now you see why I rearranged the equation above.

$${F_ar_a^2 \over m_{1_a}m_{2_a}}={F_br_b^2 \over m_{1_b}m_{2_b}}$$
I know this equation looks a bit messy, so let's simplify is a bit. The masses of the objects are the same in both situations (we're dealing with the unknown object and the Earth both times), so we can take both those guys out of the equation:

$${F_ar_a^2}={F_br_b^2}$$
Already it's looking better! We also know we want the force in situation (b) to be 2.5 times the force in situation (a). In other words, F(b) = 2.5F(a). We can plug that into the equation, and then solving becomes very easy:

$${F_ar_a^2}={2.5F_ar_b^2}$$$${r_a^2}={2.5r_b^2}$$$$r_b={r_a \over \sqrt{2.5}}$$$$r_b={6.9 \times 10^8 \over \sqrt{2.5}}$$$$r_b=4.4 \times 10^8$$
So the new distance would have to be 4.4 x 10^8 m for the force to be 2.5 times as great. Notice how the fact that the original force was 5.25 N doesn't actually matter! And this answer also makes sense because you would expect the gravitational force to become stronger as you move the objects closer together.

If you have any questions please let me know!