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Universal Gravitation and Circular Motion

Have you seen the episode of Family Guy where Brian tosses an apple at Peter and it starts orbiting him? This problem will show you just how massive you'd have to be in order to cause an apple to fall into your orbit.

Question: How much would you have to weigh in order to have an apple with mass m orbit you in a circle with a radius of r with a velocity of v?

Step one

First, I put values to those variables. For the apple's mass, I did a little research and found that the average mass of an apple is around 150 grams. I set the radius to be two meters, and I valued velocity at one meter per second.

Step two

What I did second is list out any equations I would need. This included circular motion equations and the universal gravitation equation.

Step three

Next, I established that the Fg would be equal to Fc, since factoring anything else would make things a lot more complicated. Because of this, we can say that the man is drifting through space, where no other forces can act on either him or the apple.

Step four:

I plugged in the mass, velocity, and radius into the Fc equation to figure out how powerful the centripetal force is and, since that and Fg are equal in this scenario, how strong gravity is.

Step five

After that, I set up the universal gravitation equation. I had everything but one of the masses, which is our fat old chum's mass, so I solved for that.

Step six

Now that we have the mass, which is in kilograms, we can convert it to pounds by multiplying it by 2.205.
After all that, I had a pretty big number. Like, in the tens of billions of pounds. 6.3945 x 10^10, to be exact. You can apply this approach to any problem in which you are required to find mass of an orbiting or orbited object.

Photo of my work: