From: Ronald Wong (ronwong@inreach.com)
Date: Mon Nov 13 2000 - 22:48:05 PST
Message-Id: <l03102802b63384085f61@[209.209.19.168]> Date: Mon, 13 Nov 2000 22:48:05 -0800 From: Ronald Wong <ronwong@inreach.com> Subject: Re: Why does the earth exert a large gravitational force...
Jhumki Basu said:
>We have been discussing Newton's 2nd Law in my Conceptual Physics class.
>Yesterday we discussed the fact that the earth pulls more on more massive
>objects but that they also have more inertia, so more massive objects
>accelerate due to gravity the same amount as less massive objects.
>...
There are actually three issues involved in your request for any thoughts
on this matter.
_____________________________
The first involves Newton's three Laws. He arrived at these laws after a
careful study of terrestrial phenomena. As such, they have nothing to do
with the motion in the heavens or the law of universal gravitation. They
are separate from and preceded the development of his Law of Universal
Gravitation.
According to these laws, matter resist a change in it's state of motion.
This property of matter is referred to as inertia. Mass is the measure of
inertia and, because of this property, an unbalanced force must act on an
object if there is to be a change in it's state of uniform motion. In a
good textbook, this mass is referred to as inertial mass. I'll refer to it
as m(inert).
We'll return to this concept of m(inert) later.
Now to the next issue.
_____________________________
When Newton decided to take the bold step of applying these three laws to
the motion observed in the heavens (NOT a very reasonable thing for an
educated man to do in his days), he was drawn to the conclusion that the
motion of the planets and their moons could best be explained if we
accepted the idea that a pair of attractive forces acted between two bodies
- be they planets, moons, stars, men, pencils, etc. (that there were TWO
forces acting came from his third law). One force tugged on one body and
another, equal in strength to the first, tugged on the other in the
opposite direction. The strength of either force depended on the inverse
square of the distance between the two bodies, and the product of the
amount of matter present in each.
It's really a crazy idea. So much so that we continue to test it to this
very day - over 300 years after he published his results.
In terms of our common everyday experiences, it has held up very well.
------------------------------
>A student asked why the earth exerts a greater gravitational force on more
>massive objects than on less massive objects
You can answer her/his question by pointing out that, according to Newton,
the universal law of gravity applies to everything (this idea comes from
the common sense notion that similar objects have similar properties -
whether you buy a pencil in San Francisco or Calcutta, you expect to be
able to write with it). So, what is true for the earth and an object
applies to any pair of objects - two students in the classroom for
instance.
So, point to another student and indicate to the one that asked the
question that, according to Newton, a force is acting on her/him because
of the other student. If the other student suddenly cloned an identical
twin, there would be two of him/her and, since EACH would pull on your
intrepid student with the SAME force as before when there was just one
(Again, based on the commonly held assumption that similar objects possess
similar properties), there would be twice as much force as before acting on
her/him. There is also twice as much force acting on the two students than
there was for one student (Newton's third law is still in there doing it's
thing). It's a simple proportionality. That's why "the earth exerts a
greater gravitational force on more massive objects than on less massive
objects".
The gravitational force depends on stuff. More stuff, more force.
Well, I've basically addressed the question posed by your student, Jhumki,
but if you are interested in learning more....
-----------------------------
Notice that in this context your term "massive" has to do with the amount
of matter present. It does NOT have to do with any resistance the object
has to a change in it's state of motion. That mass, m(inert), has to do
with the dynamic properties of matter and has nothing to do with gravity.
So the term "mass" here does NOT have the same meaning as the term "mass"
as used in Newton's three laws of dynamics. This one has to do with
gravitational forces and is a factor in determining it's strength. For this
reason, it is called "gravitational mass". I'll refer to it as m(grav).
Onward to issue three.
_____________________________
>Yesterday we discussed the fact that the earth pulls more on more massive
>objects but that they also have more inertia, so more massive objects
>accelerate due to gravity the same amount as less massive objects.
How apropos.
But you have the cart before the horse (you have a LOT of company in this
regard).
The fact that, at a given location, "more massive objects accelerate due to
gravity..." by "...the same amount as the less massive objects" is an
experimental fact of life. Nothing more. Nothing less.
That's what Galileo's famous experiment was all about. He confirmed that,
despite common sense, "all objects will fall to the earth at the same rate
no matter what their weight" (or words to that effect). He also showed that
the rate was constant.
There is a subtle but important assumption being made in your remarks
quoted above.
Let me explain.
When the earth pulls on the object it does so by means of a gravitational
force that depend on the size of the gravitational mass of both the earth
and the object it is pulling on.
The size of this force, F(grav), can be found using Newton's law of
universal gravitation:
F(grav) = G X m(grav-earth) X m(grav-object) / (distance between the center
of masses of these two)^2.
This force, acting on the object, is an unbalanced force, F(unbal).
According to Newton's 2nd law: F(unbal) = m(inert) X a.
Since F(grav) = F(unbal),
F(grav) = m(inert-object) X a(accel. due to gravity).
Thus:
G X m(grav-earth) X m(grav-object) / (distance between the center
of masses of these two)^2 = m(inert-object) X a(accel. due to gravity)
Here comes the important stuff:
On the left hand side of this equation, everything is a constant except
m(grav-object).
On the right hand side of this equation, there is a(accel. due to gravity).
Galileo's experiment has shown that this is also a constant.
So,
Constant(1) X m(grav-object) = m(inert-object) X Constant(2)
Or, in other words: m(grav-object) = m(inert-object) X Constant
So, although these two properties of matter represent different aspects of
matter, one gravitational and the other dynamic, they only differ by a
constant. In other words, they are proportional.
When things are "proportional" (I put in quotes because there are
conditions attached and I want to remind you of this) physicists like to
take advantage of the situation and simplify things.
You can make a proportionality into an equality by simply introducing an
appropriate conversion factor.
Here, it is simply 1 kilogram of gravitational mass equals 1 kilogram of
inertial mass. This makes the Constant equal to 1 in magnitude. As a
result, we can write:
m(grav) = m(inert).
It is important for you to remember that this is ONLY true if all the
things that we are claiming to be constant are/remain constant (remember,
for instance, that a(accel. due to gravity) was only shown to be a constant
EXPERIMENTALLY).
To get back to your quote at the beginning of this third section.
Many books like to show that the acceleration due to gravity is constant
because the unbalanced force - the objects weight - is proportional to its
mass.
F(unbal) = ma is written as F(unbal)/m = a
Since F(unbal) = Weight, F(unbal)/m = a becomes Weight/m = a
They then go on to say that when the weight goes up or down, the mass does
the same thing by the same factor. Thus the ratio, weight/m, remains
unchanged and thus the acceleration due to gravity, a, is constant
regardless of weight.
This is basically what you (and others) are saying in your remark above.
But please remember:
A. The acceleration is NOT the same because of this "fact". It
is the same because, whenever we measure the acceleration
at any given location, we find it to be independent of
mass(inert). It's an EXPERIMENTAL fact of life and we ASSUME
that this is true elsewhere. (It's this assumption that
was actually being tested when the astronaut dropped
a feather and a hammer while standing on the surface
of the moon.)
B. It's because the acceleration due to gravity is experi-
mentally proven to be the same that we can say that "as
the weight of the object increases, the objects mass(inert)
increases by the same proportion". This is what I meant
when I said you have the cart before the horse.
Remember: Weight is proportional to m(grav) and it is m(grav) that
increases proportionately with weight.
m(grav) = m(inert) only if a number of factors are constant.
There is no guarantee that the factors are or will remain constant.
We may find one day that there are conditions where the acceleration due to
gravity is NOT constant (nobody is holding their breath on this one). It
may turn out that the Universal Gravitational Constant, G, is not constant
(this one IS under serious consideration. Models have been proposed where G
changes with time for instance). It may turn out that even more fundamental
assumptions are open to question (YOU might believe that similar objects
have similar properties. It is surely consistant with common sense. But
educated people in Newton's time, and throughout the previous millennia as
far as we know, felt that common sense dictated that the laws and
principles arrived at by means of terrestrial observations did not apply to
heavenly motion. Look what happened when Newton blithely ignored that
commonly held belief).
Don't let this lengthy post lead you to believe that your students should
be aware of these fine points. That's not so. I don't think it's important
for them to know such subtle aspects of mechanics.
I think you should.
The more we know about what scientists have learned about the world we live
in and how it has changed the way they have come to look at nature and
themselves, the better we can convey to our students what science is.
A little historical perspective is important as well. I use it in my
classes to give my students a better understanding of what science is like
as a human endeavor.
Now for a related digression:
__________________________________
If you accept the idea that m(grav-object) = m(inert-object), then you can
susbstitute it into the lefthand side of the equation:
G X m(grav-earth) X m(grav-object) / (distance between the center
of masses of these two)^2 = m(inert-object) X a(accel. due to gravity)
and end up with the handy-dandy equation:
m(grav-earth) = a(accel. due to gravity) X (distance between the center
of masses of these two)^2/G
Here in San Francisco, I just substitute the mean radius of the earth for
the (distance between the center of masses of these two).
Near the end of the first semester in my conceptual physics class we do a
lab involving the pendulum. At this point, the students know that the
period of the pendulum varies with the acceleration due to gravity. I show
them the formula that allows them to calculate a(accel. due to gravity)
given the period and length of their pendulum.
After they calulate a(accel. due to gravity) using their own data, I then
walk them through the algebraic hocus-pocus given above and show them that
not only can they arrive at the world famous value for "g" (10 m/s^2) but
they can also "weigh" the earth and get it's mass. A little more plugging
away on their part (Much handholding here - most of the kids have never
used all the features available to them on their calculator) and voila! the
earth's mass.
When I've prepped them properly - well before the actual lab, they find it
really neat that it's possible to come up with the earth's mass using
simple tools and their basic algebra (the question was "How do you measure
the mass of the earth when you don't have a balance that's big enough for
the job?").
Hope this has been of some value.
ron
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