# B Action Reaction force on the earth

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1. Jul 16, 2016

### Zack K

"As you sit in your chair and study your physics (presuming that you do), the force of gravity acts downward upon your body. The reaction force to the force of the Earth pulling you downward is the force of your body pulling the Earth upwards". I read this somewhere just a bit ago and it's confusing me. How is it possible for you to be able to pull the Earth towards you with the same force? A mathematical explanation would be good also.

2. Jul 16, 2016

### Andrew Mason

Just apply Newton's Law of Universal Gravitation and Newton's Third law of motion.
The force of the earth (of mass M) on a body of mass m is:
$$F_{M on m} = m\frac{GM}{R^2}\hat{R}$$

By the third law, there must be an equal and opposite force of the body on the earth:

$$F_{m on M} = - M\frac{Gm}{R^2}\hat{R}$$

So the forces are equal in magnitude but opposite in direction.

If you consider a person falling to the earth rather than sitting in a chair, the gravitational forces are still equal and opposite, but not the accelerations. Since F = mass × acceleration the earth does not accelerate very fast toward the person. Acceleration of the earth is so small it cannot be measured. Work out the acceleration of the earth from a falling 100 kg mass using the above equation.

AM (Edited in response to A.T.'s comment below)

Last edited: Jul 17, 2016
3. Jul 17, 2016

### A.T.

If you sit in your chair, there is no acceleration, because the contact forces cancel the gravitational forces.

4. Jul 17, 2016

### Andrew Mason

Thanks. Your point is well taken, although the contact forces don't exactly cancel the gravitational forces. They still exist, of course. But the gravitational and contact forces sum to 0, so there is no acceleration. I have edited my earlier post accordingly.

AM

Last edited: Jul 18, 2016