Force = mass x acceleration (F = ma)

Date: 11/23/96 at 19:13:09
From: Cheron Frazier
Subject: Earth's acceleration?

I need help with the following problem. I don't understand it at all!

When you drop a 0.40-kg apple, Earth exerts a force on it that
accelerates it at 9.8 m/s^2 toward Earth's surface. According to
Newton's third law, the apple must exert an equal and opposite force 
on the earth. If the mass of the earth is 5.98 x 10^24 kg, what's the 
magnitude of the earth's acceleration?

Thank you!

Julian DeLorenzo

Date: 11/23/96 at 22:03:57
From: Doctor Kate
Subject: Re: Earth's acceleration?

Julian,

There are a few concepts to look over first.  Gravity is a force which 
accelerates objects toward each other. It looks a lot like an apple 
just falls to the earth, but really the earth and the apple fall 
toward each other. The earth moves toward the apple, but it moves 
such a tiny bit we can't detect it. Actually, the computer I'm typing 
at right now exerts a force on me too, but it's also very small, so I 
don't feel it. Even you, way down there in California, exert a 
gravitational force on me, way up here in North Bay, Ontario.

Now, force is always mass times acceleration (F = ma). That means 
that if a force of 2 newtons acts on something that weighs 1 kilogram, 
it will accelerate at 2 m/s, because 2 = 1 * 2 (F = m * a).  If it 
weighs 2 kilograms, it will accelerate at 1 m/s, because 2 = 2 * 1.

As the question says, Newton's third law is that a body must exert a 
force equal and opposite in direction to any force acting on it. If I 
pull your arm, your arm pulls me. If I push a door, the door pushes 
me. If you punch someone's stomach, his or her stomach 'punches' you.  
It seems strange, but why else would your hand hurt after you hit 
something?

Those are the concepts to begin with.  Look them over carefully and 
make sure you understand.  Now, for the problem.

The problem tells you that:

A: The apple has a mass of 0.40 kg and the acceleration of the apple 
is 9.8 m/s.

B: The earth has a mass of 5.98 x 10^24 kg and the acceleration of the 
earth is unknown.

C: The force of gravity exerted on the apple by the earth is the same 
as the force of gravity exerted on the earth by the apple, by Newton's 
third law.  Remember, the *force* is the same, not the acceleration.

See if you can solve it from here (please give it a try).  If you 
can't, here's a hint:

From what you know in 'A', you can find the force (F = ma).  From 
force and mass (from 'B') you can find acceleration (F = ma or 
a = F/m).  You know the forces are the same because of part 'C' above.

Now, you can probably do it from there, but I'll put my final answer 
here so you can check yours when you're done:

Acceleration of the earth = 6.56 x 10^(-25)

-Dr. Kate,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/