CHAPTER 12. FORCE,MOMENTUM AND IMPULSE 12.4
The magnitude of the attractive gravitational force between the two point masses, F is given by:
F = G
m 1 m 2
r^2(12.2)
where: G is the gravitational constant, m 1 is the mass of the first point mass, m 2 is the mass of the
second point mass and r is the distance betweenthe two point masses.
Assuming SI units, F is measured in newtons (N), m 1 and m 2 in kilograms (kg), r in meters (m), and
the constant G is approximately equalto 6 , 67 × 10 −^11 N· m^2 · kg−^2. Remember that this isa force of
attraction.
For example, consider aman of mass 80 kg standing 10 m from a woman with a mass of 65 kg.The
attractive gravitational force between them would be:
F = Gm 1 m 2
r^2
= (6, 67 × 10 −^11 N· m^2 · kg−^2 )(
(80kg)(65kg)
(10m)^2)
= 3, 47 × 10 −^9 N
If the man and woman move to 1 m apart, then the force is:
F = G
m 1 m 2
r^2
= (6, 67 × 10 −^11 N· m^2 · kg−^2 )((80kg)(65kg)
(1m)^2)
= 3, 47 × 10 −^7 N
As you can see, these forces are very small.
Now consider the gravitational force betweenthe Earth and the Moon. The mass of the Earth is
5 , 98 × 1024 kg, the mass of the Moon is 7 , 35 × 1022 kg and the Earth and Moon are 3 , 8 × 108 m apart.
The gravitational force between the Earth and Moon is:
F = G
m 1 m 2
r^2= (6, 67 × 10 −^11 N· m^2 · kg−^2 )((5, 98 × 1024 kg)(7, 35 × 1022 kg)
(0, 38 × 109 m)^2)
= 2, 03 × 1020 N
From this example you can see that the force is very large.
These two examples demonstrate that the greaterthe masses, the greater the force between them.The
1 /r^2 factor tells us that the distance between the two bodies plays a role as well. The closer two
bodies are, the strongerthe gravitational force between them is. We feel the gravitational attraction of
the Earth most at the surface since that is the closest we can get to it, butif we were in outer-space, we
would barely feel the effect of the Earth’s gravity!
Remember that
F = m· a (12.3)
which means that everyobject on Earth feels thesame gravitational acceleration! That means whether
you drop a pen or a book (from the same height), they will both take thesame length of time to hit the
ground... in fact they will be head to head for the entire fall if you dropthem at the same time.We
can show this easily byusing the two equationsabove (Equations 12.2 and 12.3). The force between
the Earth (which has themass me) and an object of mass mois
F =Gmome
r^2(12.4)
and the acceleration ofan object of mass mo(in terms of the force acting on it) is
ao=F
mo(12.5)
So we substitute equation (12.4) into Equation (12.5), and we find that
ao=Gme
r^2