Gravitation
Objective: Define the gravitational force law; calculate the magnitude of the gravitational force of one
object on another (and vice versa); understand the direction of the gravitational force of one object on
another and draw vectors for the forces on each object; calculate the approximate gravitational force of
a large body like the Earth on a small body near its surface (like me); state the principle of reciprocity
(Newton’s third law); state the superposition principle; state the principle of conservation of momentum.
Review
In the last lab, you wrote your first simulation–a computer program that modeled a physical phenomenon–in
this case, a falling ball. You now can see how powerful the momentum principle is. If you know the net force
on the object, then you can predict its momentum a short time interval later. Then after each time interval
(or time step), ∆t, you again predict its new momentum. This technique can be used to model complicated
systems and solve problems that you can’t solve by hand.
To summarize your computer program, it looked something like this:
1. define an object its initial momentum (mass x velocity)
2. define a time step, ∆t
3. define (or calculate) the net force on the object
4. predict the object’s new momentum
5. predict the object’s new position
6. step forward to a new time
7. repeat the calculations for the net force (if necessary), the new momentum, and the new position of
the object
Steps 3–7 are done within a while loop.
Note: this is calculus! Instead of solving a problem analytically (doing the integral of a function and getting
another function), we are solving the problem numerically. The reason is that it is a more powerful technique
that allows us to solve problems that cannot be solved analytically (like the infamous 3-body problem that
we’ll talk about later).
The gravitational force law
Isaac Newton understood what caused the Moon to orbit the Earth. He understood that the Earth pulled
on the Moon, and the Moon pulled on the Earth, each along a line directly between them. He understood
that it was the force of the Earth on the Moon that kept it in orbit. Also called Newton’s law of gravitation,
the gravitational force law describes the gravitational force of one particle (with mass) on another particle
(with mass). The magnitude of the gravitational force of one particle on another is
m1 m2
(1)
r2
where G is the gravitational constant, G = 6.67 × 10− 11 N m2 /kg2 . Note that the force of particle 1 on
particle 2 is equal in magnitude to the force of particle 2 on particle 1. However, the force vectors are in
opposite directions. This is true, in general, for any interaction and is called the principle of reciprocity, or
Newton’s third law. Mathematically, we can write this as
|F~g | = G
F~on 1 by 2 = −F~on 2 by 1
(2)
As an example, consider the Sun and the Earth. The Earth pulls on the Sun, and the Sun pulls back on
the Earth. Here are some questions for you to think about:
1. If the Sun pulls on the Earth and the Earth pulls on the Sun, why the forces not cancel each other out
so that the Earth and Sun will maintain constant velocities?
2. If the Earth pulls with an equal magnitude force on the Sun, why doesn’t it also orbit the Earth?
3. If the Sun and Earth are pulling each other, why don’t they collapse (i.e. collide with each other)? In
other words, what “holds” the Earth in orbit?
4. Since the principle of reciprocity is a general principle, it can be applied to other interactions. If car
A moving at 20 mi/h collides head-on with car B at rest, which car exerts a bigger force on the other?
What if both cars are moving toward each other at 20 mi/h? What if car A is actually a tractor-trailer
and car B is Honda civic?
Note also that this force is an inverse-square force, meaning that it depends on one over the distance
between the particles squared. Thus, separating two particles by twice the distance will decrease the force by
a factor of 1/4. Likewise, separating them by 3r will decrease the force by a factor of 1/9. The gravitational
force decreases dramatically with distance, and because of the small value of G, is a very small force in
general (unless the masses of the particles are very large)
The gravitational force law–approximations
The gravitational force law is correct for point particles. However, for a spherical object whose density is
uniform (or a function of radius only), we can treat it as a point particle with all of its mass at its center if
we are only considering distances larger than its radius.
If you are calculating the force of a large body on a small body that is near to the surface of the large
body, then r ≈ R, and
M
|F~on m by M | ≈ G 2 m = gm
R
(3)
where g is called the magnitude of the gravitational field and is approximately constant near the surface
of the large body. g depends on the mass of the large body and the radius of the large body. Often this
quantity, |F~on m by M |, is called weight and is written as w = mg.
In general, we can express the force on a particle of mass m as a function of this quantity called the
gravitational field ~g . The gravitational field is the net gravitational force per unit mass at a certain location
in space.
~g =
F~net,g
m
(4)
Superposition Principle
The net force on an object is the sum of all of the forces acting on that object–that’s the principle of
superposition. However, there’s an even deeper result of this principle. Each force acts on the object
independently of the other forces. In other words, that fact that the Sun exerts a force on the Moon doesn’t
affect nature of the force of the Earth on the Moon. An interaction between a pair of objects is not affected
the presence of other interactions.
Conservation of Momentum
Suppose you have two bodies interacting, yet, if you consider them as one system, there is no net external
force on the system. Then the momentum of the system is constant. Any change in momentum of one
particle must be compensated by a negative change of momentum of the other particle. That means that if
the Sun is causing the momentum of the Earth to change, the Earth is causing the momentum of the Sun
to change by the same magnitude, but opposite direction. Then why don’t we notice it? The mass of the
Sun is MUCH greater; therefore, its change in velocity is MUCH less. Mathematically, this principle can be
expressed (assuming multiple particles)
p~ = p~1 + p~2 + p~3 + · · · = constant
(5)
∆~
p = ∆~
p1 + ∆~
p2 + ∆~
p3 + · · · = 0
(6)
Application
1. Sketch the change in momentum vector for a 1000 kg car that is moving at a speed of 20 m/s in the
direction ¡1,0,0¿ and after a collision is moving at 10 m/s in the direction ¡0.709,-0.709¿. What was
the direction of the change in momentum of the car during the collision? What must be the direction
of the change in momentum of the other car, considering that the net force on the system of cars is
zero (i.e. the only interaction is the interaction of the cars)?
2. What is the magnitude of the gravitational force of your neighbor on you? What approximations did
you have to make in order to answer this question?
3. At what position between the Earth and Moon would the net force on a space probe be zero?
4. Is the total momentum of the Earth-Moon system conserved?