Download 3. Newton`s laws

Physics 107
Spring 1996
3. Newton's laws
The subject matter for the next two lectures will be Newton's laws of motion. These
three laws, together with Newton's law of universal gravitation, formed the basis of
all of physics for well over 200 years. These laws describe the motion of objects for
an extremely wide range of physical phenomena. It is not until one enters the realm
of the very small, the very fast, or the very large, that Newton's laws fail. Quantum
mechanics ( 1920), special relativity ( 1905), and general relativity ( 1915) are
needed to describe the physics in these three domains. We will briey talk about these
three theories at the end of the semester.
3.1 Force
Since all of Newton's laws involve forces in one way or another, we start by explaining
in a little more detail what we mean by the concept of \force".
Roughly speaking, a \force" can be thought of as a push or a pull that one object
exerts upon another. Like position, velocity, and acceleration, force is a vector quantity
that has both size and direction. The mathematical symbol that one uses to denote a
force is F~ .
Q: Can you give some examples of forces?
A: (i) Gravity is an example of a force. The earth exerts a gravitational force on
this tennis ball pulling it toward the center of the earth. (ii) The oor also exerts
a force on the tennis ball, stopping its downward motion and \pushing" it back
up to my hand. (iii) Friction is another example of a force. The air exerts a
frictional force on an object as it falls to the ground. The table top also exerts a
frictional force on a object as it slides across the table. Friction is a \retarding"
force that is always directed opposite to the direction of motion of the object.
An object can be subjected to more than one external force at a time. If that is the
case, we say that a \net external force" acts on the object. The net external force is
the sum of the individual external forces, in accord with the principle of superposition.
In some cases, the net external force on an object is zero. This happens if no external
force at all acts on the object, or if the sum of the individual external forces is zero. An
example of the latter possibility is a tug-of-war contest between two equally matched
teams. The two teams pull on the rope with equal but oppositely directed forces.
These two forces cancel one another, yielding a zero net external force on the rope.
3.2 Newton's 1st law
Given the above denition of force, we can now state Newton's 1st law of motion:
I: An object at rest remains at rest and an object in motion remains in motion,
moving along a straight line with constant speed, unless acted upon by a net
external force.
This law should look familiar.
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Q: What is it?
A: It is Galileo's principle of inertia with the phrase \some outside inuence" replaced by \a net external force."
The following demonstrations (table cloth and plates, string breaking apparatus, air
track with carts) should give you a avor of Newton's 1st law.
3.3 Inertia, mass, weight, and volume
The resistance of an object to any change in its state or rest or motion in a straight
line with constant speed is called \inertia." The harder it is to change the state of
motion of an object, the more inertia we say that object has. For example, a cart
lled with weights has more inertia than one which is empty. The heavy cart is harder
to start moving (and also harder to stop moving) than the lighter one. The reason
why objects behave in this way is not understood. We do not know how to explain
the principle of inertia in terms of any more fundamental concepts.
\Mass" is dened as the quantitative measure of inertia. An object with twice
as much inertia has also twice as much mass. The unit of mass in the SI (Systeme
International) system of units is the \kilogram," abbreviated kg. The unit of mass
in the English system of units is the \slug." One can convert between measurements
made in the English system and SI system of units: 1 slug = 14.6 kg.
Probably more familiar than the mass of an object is an object's \weight." The
weight of an object is dened as the net gravitational force exerted on the object.
(We will talk more about gravity in a couple of lectures.) Although they are simply
related, mass and weight are not the same.
Q: Why not?
A: Mass is a property of an object that does not depend on location. Weight, on
the other hand, does. For example, your weight as measured on the surface of
the moon is one-sixth what it is on the surface of the earth. (If you weigh 180
pounds on the earth, you would only weigh 30 pounds on the moon.)
In the SI system, the unit of weight is the \newton," abbreviated N. In the English
system, the unit of weight is the \pound," abbreviated lb. As we did for mass, we can
convert between dierent units of weight: 1 lb = 4.45 N.
The relationship between mass m and weight W on the surface of the earth is
simply
W = mg :
(1)
2
2
Here g = 32 ft=s = 9:8 m=s is the acceleration due to gravity that played a predominant role in our study of falling bodies and projectile motion. (On the surface of the
moon or some other planet, the value of g would be dierent.) This equation tells us,
for example, that an object which has twice as much mass as another object has twice
as much weight, and vice versa. By using this equation, one can show that a 1 slug
object weighs 32 lb and a 1 kg object weighs 9.8 N. A more useful relationship, which
you may already know, involves kilograms and pounds.
Q: How many pounds does a 1 kg object weigh?
A: 2.2 lb.
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Q: Can you prove this using the fact that a 1 kg object weighs 9.8 N and that 1 lb
= 4.45 N?
A: The proof is
(2)
9:8 N 4:145lbN = 2:2 lb :
Although scales in the supermarkets may be calibrated to give readings in kilograms,
what they really measure are weights, i.e., pounds.
Another property of an object that is probably more familiar than mass is \volume." But like mass and weight, mass and volume are not the same.
Q: Why not?
A: Volume is a quantitative measure of the amount of space that an object takes
up. Two objects can have the same mass (or weight), but have dierent volumes.
An example of this is a pound of feathers and a pound of rocks. They both have
the same mass (and weight), but the pound of feathers has the greater volume,
since it takes up more space than a pound of rocks.
In the SI system, the unit of volume is the \cubic meter," m3 = m m m. In the
English system, the unit of volume is the \cubic foot," ft3 = ft ft ft. Using the
conversion factor 1 m = 3:28 ft, we have 1 m3 = 35:3 ft3 . Other common units of
volume are the \liter" in the SI system and the \gallon" in the English system. They
are related by 1 gallon = 3:78 liter = :00378 m3 .
3.4 Newton's 2nd law
Newton's 1st law may be interpreted as saying that motion along a straight line with
constant speed (i.e., motion with constant velocity) is not very interesting. In fact,
motion with constant velocity is equivalent to no motion at all! Anybody who has
own in an airplane or driven in a car with constant velocity is familiar with this fact.
If you were not able to look out of the window, you would not know that you were
moving. There is no experiment whatsoever that you can do inside the airplane or
the car that would be able to tell you that you were moving with a non-zero velocity.
What is interesting, then, as far as motion is concerned are changes in velocity, i.e.,
accelerations. Newton's 1st law of motion tells you that a net external force causes
an object to accelerate; Newton's 2nd law tells you how to calculate that acceleration
in terms of the net external force and the mass:
II: A net external force acting on an object produces an acceleration of the object
which is directly proportional to the force and inversely proportional to the mass
of the object.
In words,
force :
acceleration = mass
(3)
In mathematical symbols,
F~ :
~a = m
(4)
Here F~ is the net external force acting on the object, m is the object's mass, and ~a is
the object's acceleration. Note that F~ and ~a are vector quantities (having directions
associated with them), while m is not.
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Q: What happens to the acceleration of an object if we double the size of the net
external force? What happens if we keep the net external force the same but
double the mass of the object? What happens if we leave the mass of the object
alone, but change the direction of the net external force?
A: If we double the size of the net external force, the acceleration of the object
doubles; if we double the mass of the object, the acceleration decreases by a
factor of 2. If we change the direction of the net external force, the direction of
the acceleration of the object also changes. (F~ and ~a always point in the same
direction.)
Note that by simply rearranging terms in the above equation, we can rewrite Newton's
2nd law as F~ = m~a. Moreover, if we are only interested in the magnitudes of the force
and acceleration, then we have F = ma. This is the form of Newton's 2nd law that
you nd in most physics textbooks.
3.5 Uniform circular motion
As an application of Newton's 2nd law, we will consider \uniform circular motion."
This is the motion of a body that moves in a circle at constant speed. An example is
given by swinging this tennis ball on a string. The motion of the planets around the
sun is (to a good approximation) another example of uniform circular motion.
Q: Although the tennis ball is moving with constant speed, it is not moving with
constant velocity. Why not?
A: The velocity is not constant since the direction of the tennis ball is changing as
it moves around in a circle.
As we learned a couple of lectures ago, a changing velocity means non-zero acceleration. Since acceleration is a vector quantity, it has both magnitude and direction.
Denoting the radius of the circular motion by r and the constant speed by v, one can
show (using geometry) that the magnitude of the acceleration is a = v2 =r.
Q: Can you guess in which direction the acceleration points?
A: Toward the center. The reason is that if you look at two nearby velocity vectors,
you see that they dier by a vector that points toward the center of the circle.
Using these results for the acceleration together with Newton's 2nd law, we can determine the magnitude and direction of the net external force that acts on the ball.
Q: What are they?
A: Using a = v2 =r and F = ma, we have F = mv2 =r. Since the net external force
and the acceleration point in the same direction, the net external force is also
directed toward the center of the circle. This center-seeking force is called the
\centripetal force."
Q: Does this make sense? (i.e., would you expect the force on the tennis ball to be
directed toward the center or tangent to the circle?)
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A: A center-seeking force does make sense. According to Newton's 1st law (the
principle of inertia) no force is necessary to keep the ball moving in a straight line
at constant speed. A force is needed, however, to deect the straight line motion
into a circular path. If you look in the direction of the force (i.e., toward the
center of the circle), you see what is ultimately responsible for keeping the ball
moving in its circular orbit: my hand. A similar argument can be made about the
motion of the planets. What keeps the planets moving in their (approximately)
circular orbits is the gravitational force exerted by what's at the the center of
these orbits: the sun.
Q: If I let go of the string or imagine removing the sun from the solar system, what
do you think will happen to the ball and the planets?
A: They will y o tangent to their circular orbits in straight line paths, moving
with constant speed. This is in accord with Newton's 1st law.
3.6 Newton's 3rd law
Newton's 3rd law is a statement about the nature of forces. Namely, forces are interactions between two objects: If object 1 exerts a force on object 2, then object 2 must
exert a force back on object 1. The force that object 1 exerts on object 2 is called the
\action" force. The force that object 2 exerts back on object 1 is called the \reaction"
force. In terms of action and reaction forces, Newton's 3rd law says:
III: To every action, there is an equal and opposite reaction.
In other words, you cannot exert a force on something without that something exerting
an equal and opposite force back on you.
Q: Can you give some examples of Newton's 3rd law?
A: (i) I am able to walk across the oor because as I push back on the oor with my
feet, the oor exerts an equal and opposite force propelling my feet (and body)
forward. (ii) A rie exerts a force on a bullet projecting it forward when it is
red, while the bullet exerts an equal and opposite force back on the rie. This
is the so-called \kick" of the rie. (The air track cannon demonstrates this.)
(iii) The earth exerts a force on this tennis ball as it pulls it to the ground,
while the tennis ball exerts an equal and opposite force on the earth pulling it
upward. The fact that we see the tennis ball accelerate downward and do not see
the earth accelerate upward can be explained by Newton's 2nd law: ~a = F~ =m.
Basically, the magnitudes of forces are the same in both cases, but the masses
are dierent. For a small mass like that of the tennis ball, the acceleration is
relatively large. But for a large mass like that of the earth, the acceleration is
extremely small. (If you plug in the appropriate numbers for the mass of the
tennis ball and the mass of the earth, you nd that the accelerations dier by
a factor of 1025 . This is a number with 25 zeroes after it! No wonder why we
don't see the earth accelerating upward.)
3.7 Summary
Newton's three laws are summarized below:
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I: An object at rest remains at rest and an object in motion remains in motion,
moving along a straight line with constant speed, unless acted upon by a net
external force.
II: A net external force acting on an object produces an acceleration of the object
which is directly proportional to the force and inversely proportional to the mass
of the object: ~a = F~ =m.
III: To every action, there is an equal and opposite reaction.
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