Download Lecture # 5, June 13

Lecture 4.1
Force, Mass, Newton's Laws
Throughout this semester we have been talking about Classical Mechanics which
studies motion of objects at every-day scale. Classical mechanics can be subdivided into
several areas. The part of classical mechanics which studies classification and
comparison of motion and tries to answer the question: “How does the motion occur?” is
called kinematics. We have already quite succeeded in our study of kinematics. Now we
are able to describe different types of motion including multi-dimensional motion and
even rotation. But we cannot study motion without understanding its reasons. This is why
mechanics is not limited to kinematics only. The other part of classical mechanics which
studies causes of motion is called dynamics. Dynamics is based on the three Newton’s
laws. General discussion of these laws is the subject of today’s lecture. These are
universal laws. All macroscopic objects surrounding us in our every day life obey these
laws. It does not matter if we are talking about motion of the Moon or about motion of an
apple falling from the tree, the general principles of motion are the same.
What, in your opinion, causes things to move?
Answering the question what is the reason of motion, you might notice that every
time when a body changes its velocity, it interacts with some other objects. For instance,
if we are talking about a motion of the ball during any sport game, it changes its velocity,
because it was pushed or cached by a player or because it was affected by gravitational
field of the earth. The same occurs with King Kong falling from the skyscraper. He is
affected by the earth's gravity. Of course, the nature of these interactions differs. In the
case of the ball kicked by a football player, we have an example of the short-range
interaction. It occurs only during a short period of time of the actual contact between the
ball and the player’s foot. In the case of King Kong falling from the Empire State
Building, we have an example of long range interaction, where King Kong is constantly
affected by the earth’s gravity. However, there is no actual contact between King Kong
and the earth until he hits the ground. In this case we are talking about interactions by
means of some field (gravitational field of the earth in this example).
Exercise: Think about other examples of short and long-range interactions.
Interaction with other objects causes physical bodies to change their velocity. This
interaction is associated with force, which acts on a body. The relationship between force
and acceleration is the key subject of Newtonian mechanics, named after Isaac Newton
(1642-1727), who first formulated this relationship qualitatively as well as quantitatively
in the form of the Newton's laws of motion.
Even before talking about Newtonian mechanics, I would like to emphasize once
again the same aspect which we have already discussed during the first lecture.
Newtonian mechanics provides essentially classical description of the surrounding world,
which is only valid at certain scales. Even though it seems reasonable for us, based on our
everyday experience, to think about everything as a perfect Newtonian machine, we
should not extrapolate this picture on entire world. One can not use Newtonian mechanics
at very small atomic scale, where quantum mechanics should be used, neither one can use
it at the scale of universe, where General Theory of Relativity is applicable. Also it is not
possible to use these laws for the motion at very high (comparable to speed of light)
speed. In this case Special Theory of Relativity has to be used. All other situations,
known as everyday life, fit quite well in the framework of Newtonian mechanics. This is
why, those laws are so important.
1. Newton's first law
We already said that people use to extrapolate the "common sense" to situations
where it does not work. For instance, it is a common sense that a moving body will
eventually stop, if you do not apply some special force to keep it moving.
Do you think that this is true?
For many centuries people thought that the only natural state for a body is to be at rest, if
no forces are acting on it. This misunderstanding was even supported by the teachings of
Aristotle, who put it in the basis of his natural philosophy. But let us look at this issue
closer. Suppose we have a hockey puck placed on the floor. If we push this puck across
the floor, it will move for some time but eventually it will stop. Now if we do the same
experiment not on the floor but on ice and if we push it in exact same way, it will move
for much longer. As you can see, is not necessary to push the puck all the time during the
motion for it to be moving. Moreover the same original push gives different resulting
time and distance of motion at different conditions. If we somehow could improve those
conditions making the puck and ice smoother, it will travel for even longer distance. In
ideal case, if we could only remove the ice surface at all it will move forever. So the true
reason why the puck is stopping is not because it was not pushed or pulled, but because it
was affected by friction and air resistance. This means that for a body to move with
constant velocity is the same "natural state" as if it was not moving at all. This is the
content of the Newton's first law, which states that
A body remains in a state of rest or constant velocity (zero acceleration) when it is
not affected by other bodies.
So, if a body is at rest it will stay at rest, if it is moving with constant velocity it
will continue doing so until influenced by other objects. Actually there is no difference
between the two situations of a body at rest or moving with constant velocity. Indeed, the
question of velocity depends on the reference frame. The object may have completely
different velocities in two different reference frames. For instance, if you are a passenger
on the airplane and you are siting in your chair all the time during the flight, then your
velocity is zero with respect to the plane, while it is definitely not zero with respect to the
ground. On the other hand you can move through the plane during the flight. In this case
your velocity will not be zero with respect to the plane, but it still will be much larger
with respect to the ground. If this plane flies with constant speed and the distance of
flight is not too large (so we can ignore the earth's curvature) and if all the windows are
closed then there is no way for a person in the plane to know how fast this plane is
moving or if it moves at all. The only time when you can “feel” that the plane is flying is
when it changes altitude or experiences bumps. But in such a case there are forces acting
on it. This explains why some reference frames are called inertial reference frames.
Simply speaking, these are the reference frames where Newton's first law is working. It is
also often called the law of inertia. By inertia we mean a property of the body to keep its
velocity constant until it is affected by an external force.
2. Newton's second law
We have already mentioned word "force" several times, but we have never defined
this term. Now it is the right time to do so. The force is the reason for the body to change
its velocity or in other words to obtain acceleration.
Think what are the main characteristics of force?
It can be characterized in several ways. First, since the force acts on a body, it is applied
to this body and has a certain point of application. Second, it has the direction in which it
is applied and, third, it has the magnitude which shows how strong it is.
Physical quantities, having both magnitude and direction, are called vectors. Based
on the properties of force, we can conclude that force is a vector. This means that forces
must obey the same laws of addition as all other vectors. For instance, if there are several
forces acting on a same body, one can define the resultant force or the net force acting on
the body as a vector sum of all acting forces:


Fnet   F
(4.1.1)
Returning back to the example about a puck on the floor, there are several forces
acting on it. These include the pushing force, which we have applied in order to move it,
the friction force from the floor or from the ice, the air resistance, the gravitational force,
and the normal force of the floor. However, adjusting the force which we have applied to
this puck, we can make it move with constant velocity, like no force is acting on it at all.
This does not mean that there are no forces. That only means that the net force acting on
the puck is zero. So, we can reformulate Newton's first law in a different way as: The
body's velocity stays constant if the net force acting on the body is zero.
Now let us notice that even if we apply a very same force it may affect different
objects differently. Indeed we can push light hockey puck on the floor to make it move,
but we can also push with the very same effort on a heavy box full of books and most
probably it will not move at all. This means that the body's acceleration depends not only
on the applied net force, but on some other property of the body as well. This other
property is mass of the body. To see that, one can conduct an experiment by applying the
same force to two bodies with different masses and measure the resulting acceleration of
these bodies. However, if we wish to perform this experiment, we have to give the exact
definitions of physical quantities, which we are going to measure, such as force. We
know that to measure means to compare with standard, so we shall use something as a
standard of force. For instance we can use a spring scale. Indeed the harder you push or
pull on a spring the more compressed or stretched it will be. Compression or stretching of
the spring is easy to measure. So, all you have to do is to calibrate your spring scale for
measurements of force. As long as the force is not too large, there is a linear relationship
between the change of the spring’s length and force acting on the spring.
If we now conduct an experiment, it will show that the resulting acceleration is
proportional to the applied net force
 
a  Fnet
(4.1.2)
and inverse proportional to the mass
a
1
.
m
(4.1.3)
Arrows above the force and acceleration in equation 4.1.2 show, that they both are
vectors having the same direction.
In fact, this experiment is the only way to measure mass. It can be done by applying
the same force to different objects and measuring their accelerations. This experiment
shows that mass is the intrinsic characteristic of a body which comes to existence with
the body itself. To measure mass we can perform the above experiment first applying the
force to the mass standard and then applying the same force to the unknown mass. If one
compares the resulting accelerations, he/she can conclude how many times the mass of
the object is different compared to that of the standard of mass. Since, the direction of the
resulting acceleration is the same as of the applied net force, the mass is a scalar.
Moreover, mass is an additive quantity, which means that, if we apply the same force to
two masses combined together, we will still have equation 4.1.3 valid with mass as the
sum of the two masses.
Now we can formulate Newton's second law: The net force acting on the body is
equal to the product of the body's mass and the acceleration produced by this force or


Fnet  ma .
(4.1.4)
Applying this equation one has to remember that we are talking about the net force acting
on the body not just about any force. We also have to remember that this net force acts on
that body, not just any body in the system.
Equation 4.1.4 also tells us that if the net force acting on the body is zero, then the
resulting acceleration is also going to be zero, which brings us back to the statement of
the Newton's first law. In this case, we say that the body is in equilibrium and all the
forces acting on it are in balance. In some particular reference frame we can distinguish
between dynamic equilibrium, when the body moves with constant speed, and static
equilibrium, when body is at rest.
Equation 4.1.4 also helps us to define the SI unit of force, which is
1 Newton (N )= 1kg  1m s2 ,
the force providing acceleration of one meter per second squared acting on the mass of
one kilogram. There are other units of force in different systems of units. For instance,
British unit of force is Pound (it is not unit of mass as people say sometimes).
Talking about forces in nature we can distinguish between several main types of
forces or interactions. Those include: Gravitational interaction, Electromagnetic
interaction, Strong nuclear interaction and Weak nuclear interaction. The greatest task of
physics is to unify all these interactions under the same basic law.
Since Newton's second law involves the net force acting on the body, in order to
solve any problem using this law, first of all one has to draw a free-body diagram. This
diagram shall include all the forces acting on the body, but it shall not include any forces
acting on other bodies. It also shall include the net force, which comes as a result of the
vector summation of all the forces acting on the body. One also needs to include a
coordinate system, since the most effective way to add vectors (in this case forces) is by
adding their components. So, the difficulty level of the problem very much depends on
the correct choice of coordinate system.
The Newton's second law may be applied not just to one body but to a system of
bodies considered as a whole. In this case you will only need to include external forces
acting on this system. But you should not include any internal force acting between the
different parts of this system, since they cannot provide acceleration to the entire system.
For instance, if you are considering a train pushed by a locomotive as the whole system,
you should take into account the pushing force of the locomotive as an external force. But
you should not include the internal pulling forces between the different carts in the train,
since you are considering them as all moving together.
Applying equation 4.1.4 one has to remember that this equation is a vector
equation, so it can be presented in terms of vector components as
Fnet , x  ma x ,
Fnet , y  ma y ,
(4.1.5)
Fnet ,z  maz .
So, one can see that the component of acceleration along the certain coordinate axis is
only caused by the net force’s component along the same axis.
3. Newton's third law
We just have considered several examples of interacting bodies, such as the floor
and the box on the floor or the train consisting of several carts. We also said that one
should not include forces acting in between the carts to the Newton's second law equation
when considering the train as a whole, since these are internal forces. Now let us take a
closer look at these forces acting between the different objects. If we have two carts in
the train, let us say A and B, and the train moves from the left to the right, having cart A

before cart B, then cart A pulls on cart B with force FAB directed from the left to the

right. At the same time cart B pulls on cart A with force FBA directed from the right to the
left. The similar situation took place in the previous examples, since the box was pulling
on the floor with its weight, while the floor was pulling on the box in the opposite
direction with normal force. Newton's third law has to do with these pairs of forces and it
states that
When bodies interact, the forces on the bodies from each other are always equal in
magnitude in opposite in direction or


FAB   FBA .
(4.1.6)
Note that the statement of the third law is not trivial and cannot be reduced to the special
case of the second law. This is because the second law tells us about forces acting on the
same object, while Newton's third law tells us about the forces applied to different


objects. Force FAB acts on object B, while force FBA acts on object A and also called the

reaction force to FAB .
Let us continue with this example about the train. We shall call the force of the

locomotive acting on the first car A from left to right to be F and we will neglect friction
and air resistance, so the Newton's second law for the first cart will be
  
mAa  F  FBA ,
mAa  F  FBA ,
where we have chosen the direction from the left to the right as positive direction. The
Newton's second law for the second cart B is
 
mB a  FAB ,
mB a  FAB.
We can add these two equations, which gives
mA a  mB a  F  FBA  FAB ,
(mA  mB )a  F ,
a
F
.
mA  mB

In this equation we have used Newton's third law, which allowed us to cancel forces FAB

and FBA , so we ended up with the Newton's second law for the entire train and found its
acceleration. If the third law was not true, one could not apply the second law to each of
the carts and to the entire train at the same time.