|
AN INTRODUCTION TO NEWTONIAN
MECHANICS by Edward Kluk Dickinson State University, Dickinson ND |
Definition of gravitational mass and how to compare
forces
Newtonian calculus
Formulation of the second law and
problem of inertial mass
p(t) = mI v(t) .
Newton proposed that a rate of change of momentum Dp(t)/Dt
should be equal to the force F(t) acting upon the body which
we can write as
Dp(t)/Dt = F(t) .
This formula represents Newton second law in its original formulation. In
most of practical cases an inertial mass of the body is constant then it
does not depend on time. Consequently
Dp(t) = p(t) - p(t - Dt) = mI v(t) - mI v(t -
Dt) = mI [v(t) - v(t - Dt)] = mI Dv(t)
which leads to more popular form of Newton second law
mI Dv(t)/Dt = F(t) or
mI a(t) = F(t)
where a(t) is a rate of change of velocity or a body's
acceleration. It is very important to realize that the second law does not
define force or inertial mass. But it makes possible to predict the motion
if the force, inertial mass and some initial conditions for this motion are
known. Another important and unsolved yet problem represents inertial mass.
Formally at this stage we do not know how to measure it.
Solution of the inertial mass
problem
The verifying "experiment"
the block of mass mB which can slide on the frictionless
horizontal surface
the hanger of mass mH connected with the block with
help of massless string running through the pulley
the pulley which will not rotate because there is not friction between the
pulley and the string.
We will investigate motion of the system containing the block, hanger and
string. A total mass of the system m = mB + mH
will be kept constant. If we let it go, the system will
move with a constant acceleration caused by a constant gravitational
force F acting on the hanger. Knowing the hanger mass we can
find this force as F = mH g . Remember you are on
the surface of the strange planet, not Earth. Timing the block or hanger
every two meters and making the graph of the travelled distance versus
t2 we can find the system acceleration exactly the
same way as we did it for the free fall motion. Remember to include into
your graph the point t = 0 with the travelled distance
also equal to zero.
Please save all the data related to this "experiment" because
they will be used again in conjunction with mechanical energy problems.
Short epilogue
Evaluation
the objectives of this lesson are fully achieved. If you have doubts try
to read it once more concentrating on them, but do not try to memorize this
text. Physics is not about memorizing, it is about understanding.
This time we are facing a very formidable
task to analyse and truly understand Newton laws of motion. In a process
of doing it a few new ideas and related to them operational definitions must
be established. These definitions will enable us to measure magnitudes which
are characterizing quantitatively mentioned ideas.
Long before Newton people were measuring,
with help of simple two plate symmetric or spring balances, gravity
forces that are pulling bodies down toward the center of Earth . Such
measurements were mostly done for commercial purposes to compare amounts
of matter, scientifically called masses, with certain
standards. In 1790 Paris Academy of Science established a standard of mass
called 1 kilogram (1 kg) which is still used in
International System of
Units (SI). If a certain body placed on one plate of the balance needs
to be equilibrated by two copies of 1 kg standard placed on the other plate,
its mass is 2 kg. Moreover, the gravity force acting on this body is
two times greater than the gravity force acting on 1 kg standard mass. In
every day language these forces are called weights and the measuring
process is called weighing. In this scheme masses and weights are
evidently proportional. But exact relation between them (a coefficient of
proportionality) is still missing. There is one more interesting thing. If
an object is moved from the surface of Earth to the surface of another planet
its weight will be different because a gravity force exerted by the planet
on the object will be different. Its mass, however, will stay the same. For
example, 2 kg object on the other planet must still be equilibrated on the
two plate balance by two copies of 1 kg mass standard. This is why the idea
of mass is more generic than the idea of weight. But comparing masses of
other bodies with use of balances we are always employing gravitational forces.
This is why such masses should be called gravitational masses. Later
on you will learn why we do not use the added adjective too often. Now, try
to figure out on your own why all these ideas are still working if a good
spring balance is used. The good spring balance must have the spring which
elongates proportionally to the applied force. Notice that using springs
we can compare magnitudes of other forces to magnitudes of gravitational
forces. For example, attach one end of an exercise spring to the ceiling
and stretch it 0.2 m by pulling it down with your arm. Now, instead of stretching
it yourself, hang on it such amount of mass which will stretch it 0.2 m.
The gravity force acting on this mass must than be equal to the force you
were exerting on the spring.
Our investigation of rectilinear motion
on a horizontal frictionless plane let us conclude that if a body is left
alone (no a net force is acting upon the body) it moves with constant velocity.
If the body stays at rest this constant velocity is equal to zero . So far
we have learned how to compare forces and we can measure their relative
magnitudes. We also know that forces are influencing body's motion. But we
do not know any quantitative relations between the body's motion and applied
forces. This situation will change as soon as Newton second law is introduced.
A meaningfull introduction of it, however, is not possible without a little
bit more sophisticated algebra. Limiting our discussion to 1D motion of a
point like body along x axis we can mark a current position
of the body at time t as x(t). Traditionally
a small change of any variable is denoted with help of capital greek letter
"delta" which looks like an equilateral triangle. Unfortunately HTML in its
current form does not allow this kind of letter. Therefore it will be replaced
by D. Consequently Dt will represent a small
change of time and Dx(t) = x(t) - x(t - Dt) a small change
of x that takes place in the time interval (t, t - Dt). The
same kind of notation will be applied to other variables which are dependent
on time. Right now it is not difficult to notice that Dx(t) /
Dt represents a rate of change of body's position or a body's
velocity. Assuming Dt positive, if
Dx(t) is positive the velocity is positive too, and the
body at this particular instant t moves in the
positive direction of x axis. If Dx(t)
is negative then the velocity is negative too and body moves in the negative
direction of x axis. Understanding of this kind of math
is very important for our further discussion.
As we already have noticed a change
of body's velocity demands application of a force. More massive is the body
more force is needed to induce the same change in its velocity. The resistance
of the body against a change of its velocity is called the body's inertia.
Notice that the body's inertia may have nothing to do with body's gravitational
mass. The last demonstrates itself and can be measured only if the body is
under influence of a gravitational force. The body's inertia demonstrates
itself always in this senese that body's velocity cannot be changed without
application of a force or inducting a change of its inertia. For sake
of simplicity we will discuss here only bodies with constant inertia. Following
Newton let introduce another idea which he called an amount of body's
motion. Now it is known as a body's momentum
p(t) and defined as a product of body's inertial mass
mI and its velocity v(t). Thus we
may write
Historically verification of Newton
second law took many years because of lack of proper technologies and enough
advanced mathematics. Remember that to formulate the second law Newton was
forced to invent a calculus similar to what we are using in our lectures.
Latter developed advanced mathematics helped to confront Newtons second law
with astronomical data and confirm its correctness. In our verification we
will rely mostly on unusual properties of our fictitious planet like its
very low gravity acceleration and lack of friction. The second law describes
correctly motion of a body if a net force acting upon this body is equal
to zero. A zero force implies a zero acceleration which in turn implies a
constant velocity. In a case of free fall motions experiments show
a constant and equal for everybody acceleration g. On Earth
g is about 9.8 m / s2 and on our strange
planet about 0.01 m / s2. In both cases the
second law implies a constant force F = mI g.
On the other hand we know that this case F is a gravitational
force and then it must be proportional to the gravitational mass of the body.
Consequently for everybody inertial mass is proportional to its gravitational
mass. Because there are not other constrains on inertial mass it is very
convenient to choose for a proportionality coefficient the plain number
1. Thus both masses will have the same units and the same values,
but not necessary the same nature. Additionally, from now on we will skip
subscript I used with inertial mass.
Now we know enough about the second
law to verify it "experimentally". The "experimental" set contains (see the
applet above):
If you are already convinced about constant acceleration of the system the
"experiment" could be simplified. Just measuring only a flight time
of the block from 0 m to 8 m and using an appropriate formula we could
calculate an acceleration. Please avoid this simplification and time the
block or hanger every two meters because the collected data will be used
again for the study of system mechanical energy.
For each consecutive run a portion of mass should be moved from the block
to the hanger. It will increase the force F running the system
, without changing the total mass of the system. When enough data pairs force
- acceleration are collected than a graph force vs. acceleration
can be made. As the total mass of the system was kept constant, then according
to the second law this graph should be a straight line through (0,0) point
and its slope should be equal to the total mass of the system. If you understand
the idea of this "experiment " go ahead find a partner and make it. It is
easier for two people to make the "experiment". One person shall observe
the block or hanger and signal when to read the time, whereas the other person
shall read and record the time. Make all graphing and calculations needed
to verify the second law. Compare the inertial mass of the system recovered
from the slope on the graph force vs. acceleration with the real mass of
the system. They should not differ more than 5%. Any discrepancy between
them is related to both experimental errors in time and distance measurements
and imperfectness of graphing.
Laws formulated by scientists have
their limitations. That means they apply only under certain conditions and
if these conditions are not met they do not work. To make them work with
less restrictive conditions they have to be generalized. If Newton second
law were exact for bodies on and around Earth we would not have stationary
satellites nor hurricanes. Shortly you will learn about the second law
limitations.
If at this point you can solve the
following problems:
Last update: Jan 10, 1997 | E - mail to Edward Kluk |
Copyright (c) 1996 Edward Kluk |