12.2 - G and g
Newton’s law of gravity includes the gravitational constant G.
In this section, we discuss the relationship between G and g, the rate of freefall
acceleration in a vacuum near the Earth’s surface.
The value of g used in this textbook is 9.80 m/s^2 , an average value that varies slightly
by location on the Earth. Both a 10-kg object and a 100-kg object will accelerate toward
the ground at 9.80 m/s^2. The rate of freefall acceleration does not vary with mass.
Newton’s law of gravitation, however, states that the Earth exerts a stronger
gravitational force on the more massive object. If the force on the more massive object
is greater, why does gravity cause both objects to accelerate at the same rate?
The answer becomes clear when Newton’s second law of motion, F=ma, is applied.
The acceleration of an object is proportional to the force acting on it, divided by the
object’s mass. The Earth exerts ten times the force on the 100-kg object that it does on
the 10-kg object. But that tenfold greater force is acting on an object with a mass ten
times greater, meaning the object has ten times more resistance to acceleration. The
result is that the mass term cancels out and both objects accelerate toward the center
of the Earth at the same rate, g.
If the gravitational constant G and the Earth’s mass and radius are known, then the
accelerationg of an object at the Earth’s surface can be calculated. We show this
calculation in the following steps. The distance r used below is the average distance
from the surface to the center of the Earth, that is, the Earth’s average radius. We treat
the Earth as a particle, acting as though all of its mass is at its center.
Variables
Strategy
- Set the expressions for force from Newton’s second law and his law of gravitation
equal to each other. - Solve for the acceleration of the object, and evaluate it using known values for
the other quantities in the equation.
Physics principles and equations
We will use Newton’s second law of motion and his law of gravitation. In this case, the
acceleration is g.
Step-by-step derivation
Here we use two of Newton’s laws, his second law (F = ma) and his law of gravitation (F = GMm/r^2 ). We use g for the acceleration instead
ofa, because they are equal. We set the right sides of the two equations equal and solve for g.
G and g
G= gravitational constant everywhere in
universe
g = freefall acceleration at Earth’s
surfaceg = free fall acceleration at Earth’s
surface
G = gravitational constant
M = mass of Earth
r = distance to center of Earth
gravitational constant G = 6.67×10í^11 N·m^2 /kg^2
mass of the Earth M = 5.97×10^24 kg
mass of object m
Earth-object distance r = 6.38×10^6 m
acceleration of object g