Mechanics
Isaac Newton
25 December 1642 - 20 March 1727, Julian calendar
4 January 1643 - 31 March 1727, Gregorian calendar
• Books
1687: Philosophae Naturalis Principia Mathematica, or
Mathematical Principles of Natural Philosophy
The Principia is justly regarded as one of the most important
works in the history of science.
• Newton’s Laws of Motion
(1) Law of Inertia: (Galileo) A body free of forces moves in a
straight line with constant speed.
(2) A body of mass m kilograms under a force F~ Newtons has
an acceleration ~a meter/second/second in the direction of the
force with F~ = m~a. The arrow over F~ , or ~a, means that it is
a vector, so its direction is important. For example, a force of
10 Newtons acting on a 2 kg body gives it an acceleration of 5
m/s2 .
(3) “To every action there is always an equal and opposite
reaction: or the forces of two bodies on each other are always
equal and are directed in opposite directions”. In other words,
whenever a first body exerts a force F~1 upon 2 upon a second
body, the second body exerts a force F~2 upon 1 upon the first
body, such that F~1 upon 2 and F~2 upon 1 are equal in magnitude
and opposite in direction.
Conservation of Linear and Angular Momentum: A particle of mass m and velocity ~v has linear momentum p~ = m~v .
The law of inertia says that p~ is constant for a particle moving free of forces. The interesting thing is that a system of N
particles has total momentum p~total = m1~v1 + m2~v2 + m3~v3 +
...mN ~vN . As these particles move, or collide, exerting forces
on each other, the total momentum is constant if there are no
external forces acting on the system. An example of this is
the recoil of a rifle in the figure to the left below
M = mass of rifle
m=mass of bullet
M much greater
than m
Rifle and bullet at rest, with zero
momentum
m
M
V
v
Final momentum = 0, so MV = mv
V much smaller than v
The rifle and bullet initially have zero velocities. After firing
the forward momentum mv of the bullet ie equal and opposite
to the backwards momentum M V of the rifle, so M V = mv,
m
or V = M
v, with V much smaller than v because M is much
bigger than m.
Why is it that when a ballerina pulls in her arms, or a diver
tucks in his knees, they spin faster? Rotational motion and angular momentum are more complicated. In many simple cases
a body rotating with angular velocity ω
~ has angular momen~ = I~
tum L
ω where I is the moment of inertia of the body,
which depends on the mass, size, and shape of the body. For
2
example, a sphere of mass M and radius R has I = 2M5R , but
2
a cylinder has I = M2R . A body spinning free of external
~ constant. Say that Linitil = Lfinal , theretorques moves with L
fore Iinitial ωinitial = Ifinal ωfinal , then ωfinal = IIinitial
ωinitial . The
final
interesting thing is that if If inal becomes smaller than Iinitial ,
such as when a ballerina pulls in her arms, or a diver tucks in
his knees, ωfinal becomes bigger tham ωinitial and so the body
spins faster.
Acceleration: Galileo was the first to understand acceleration. Acceleration is the rate of change of velocity. A car that
starts from rest with a constant acceleration a = 10 mph/second
reaches a velocity v = 60 mph in a time t = 6 s. That is,
v = at.
The distance covered is d = vaverage t =
1 2
at .
2
Galileo discovered that if air resistance is negligible, all bodies
near the Earth fall with the same acceleration g = 9.8m/s/s,
or g = 9.8 m/s2 , no matter what is the weight of the object.
That is, starting from rest, after t seconds, the speed of the
object is
v = gt,
and the distance it falls is d =
1 2
gt .
2
So, after t = 1 second the speed is v = 9.8m/s2 × 1s = 9.8 m/s,
and the distance traveled is d = (1/2) × 9.8m/s2 × 1s = 4.9 m.
After t = 3 sec the speed is v = 9.8m/s2 × 3s = 29.4 m/s, and
the distance is d = (1/2) × 9.8m/s2 × (3s)2 = 44.1 m, and so
on.
Circular motion (Huygens): A body moving with speed
v in a circle of radius r has an acceleration with magnitude
a = v 2 /r directed toward the center of the circle, even if it is
going around with constant speed. The acceleration is due to
the fact that the velocity vector ~v is changing direction as it
goes in a circle.
v
m
a
R
• Newtonian Gravity: Gravity has long range but is the
weakest of the four fundamental interactions. The weight of
an object is the force of the Earth’s gravity upon the object.
Newton explained his law of gravity saying that any two bodies
with masses of m1 kg and m2 kg separated by a distance of r
meters attract each other.
F= a
m
Law of Motion
m
a
F
m1
m2
Force of gravity
F
F
r
The magnitude of the force of attraction is
Fgravity =
Gm1 m2
where G = 6.67×10−11 Newton · m2 /kg2 .
2
r
With this force the Sun attracts the planets and keeps them
in orbit, or the Earth attracts an apple and accelerates it with
g. From this we explain the acceleration of gravity g, because
a body of mass m with acceleration ~g must be under a force
F~ = m~g , but the Earth of mass ME attracts a body of mass m
on the surface of the Earth, at a distance RE from the center
of the Earth, with a force of magnitude
F =
GME m
GME m
GME
,
so
F
=
mg
=
,
or
g
=
2
2 .
r2
RE
RE
The constant G is very small, and Newton did not know its
value, but he knew g. When Cavendish measured G in 1798,
the experiment was called weighing the Earth, because then,
with RE ≈ 6.4 × 106 m,
2
gRE
9.8 m/s2 (6.4 × 106 m)2
ME =
=
≈ 6 × 1024 kg.
−11
2
2
G
6.67 × 10
Newton · m /kg
Work and Energy
Energy is the ability to do work. Energy is conserved. There
are many types of energy. The most important is energy of
motion, or Kinetic energy. When a body moves a distance d
in the same direction of a total force F that acts on it during
the entire motion, we say that the force has done work
W = F d, and the body acquires kinetic energy KE =
1
mv 2 ,
2
where m is the mass, and v is the speed of the body. The unit
of work and energy is the Joule. In order to be able to do work
one must have a supply of energy to ”pay” for it. For example
a force of 5 Newtons acting over a distance of 3 meters does
15 Joules of work. If there is there is no movement, there is no
work. So, for example, the columns that are supporting the roof
are not lifting the roof and therefore are not doing any work
and do not need a supply of energy. A body of mass m = 2 kg
moving with speed v = 10 meter/second, has a kinetic energy
of 50 Joules. A Joule is the same as a Watt·second. So a
60 Watt bulb, running for 10 seconds, consumes 600 Joules
of energy. The kilowatt-hour is also an energy unit equal to
3,600,000 Joules. In your electric bill you pay about 10 cents
for a kilowatt-hour of electrical energy.
Potential Energy
In order for an ”agent”, such as a person or a machine, to slowly
lift (little kinetic energy) a body of mass m kilograms by a distance h it is necessary for the agent to exert an upward force to
overcome the weight of the body, which is W eight = mg, with
g = 9.8 m/sec/sec. Therefore the work needed to lift the body
is W ork = F orce × distance = mgh. For example if m = 5
kg, and h = 2 meters, then =W ork = 5kg × 9.8m/sec/sec ×
2meters = 98 Joules. This work is not manifested as kinetic
energy of the body, but rather as Gravitational Potential
energy, which is stored in the gravitational field and can be
recovered, such as by letting the body drop. Then that energy
is converted to kinetic energy of the body. In general, potential
energy is the energy a body can have by virtue of its position
in a field of force, such as gravity, electrical forces or nuclear
forces. For example, a body at a height h has a gravitational
potential energy P E = mgh. As a body falls and gathers
speed, potential energy is converted to kinetic energy. Chemical energy is mostly due to the potential energy of electrical
forces in atoms.
Heat and Energy
The internal energy of a body is the sum of the kinetic and
potential energies of the random motions of its atoms. Temperature is a measure of the internal energy of a body. Recall
that the Fahrenheit temperature TF is related to Celsius (or
Centigrade) temperatute TC by the formula TF = (9/5)TC +32.
A more important temperature scale is the absolute (or Kelvin
) temperature TA which is related to TC by TA = TC + 273.16.
The average kinetic temperature of a molecule of a body is
related to TA by the formula
KEaverage =
1
3
mv 2 = kTA
2
2
where k = 1.38 × 10−23 Joule/(degree Kelvin). Heat is the
flow energy by virtue of a temperature difference, that either
increases or decreases the internal energy of a body. The Calorie is an energy unit used to measure heat. The food calorie,
or diet calorie, also called Calorie, or kilocalorie=4,186 Joules.
Conservation of Energy
The important principle of the conservation of energy, which
says that energy is neither created nor destroyed, but only
transformed, was not formulated until it was understood that
heat is a form of energy. For example a book sliding on a table
against friction slows down and stops. What happened to its
kinetic energy, which seems to have disappeared? Well, it was
converted to internal energy of the table and the book, which
are both a little warmer than before. This can be checked
because we know that the heat needed to raise the temperature of a body of mass m from an initial temperature Ti to
a final temperature Tf is Heat energy = cm (TF − Ti ), where
cm = ch m is the heat capacity of the body, and ch is the
specific heat capacity per kilogram. For example, for water
ch = 1 Calorie/kilogram, or 4,186 Joules/kilogram, which is
large compared to most materials.
POWER
The unit of power is the Watt. Power is the rate of doing work,
or consuming energy. That is, if work W (in Joules) is done
(or energy consumed) during a time t (in seconds), the power
P is the work divided by the time, or
P =
W
Joule
in units of 1
= Watt, or 1Joule = 1Watt · second
t
second
For example a 60 Watt lightbulb running for 3 seconds consumes W = 60 Watt × 3 seconds = 180 Joules.
The four Fundamental Interactions
• Gravity: It acts upon mass-energy, which is conserved. It is
the weakest, and long range. It is an attractive force. It keeps
planets, stars, galaxies together. general relativity explains it
in terms of a curvature of space-time. Quantum gravity is not
yet a successful theory.
• The Weak Interaction: It acts upon the three ”flavors” of
lepton number. The electron, muon, and τ lepton flavor numbers, are separately conserved. The electron, muon, and the
τ , as well as their associated electron neutrino, muon neutrino,
and τ neutrino have lepton number +1 (each of its kind). Their
antiparticles have -1 lepton numbers of each kind. It is weak
and very short range (nuclear dimensions). It is important in
radioactive decay, for example,
neutron → proton + electron + electron antineutrino. It is
mediated by an exchange of W + , W − , and Z 0 mesons.
• Electromagnetic Interaction: It acts upon electric charge,
which is conserved. The proton has charge +e = 1.6 × 10−19
Coulombs and the electron has −e. Opposite charges repel
and like charges attract each other. It is strong and long range
enough that we have built a great technology using it. It keeps
atoms together. It is mediated by an exchange of photons.
• The Strong Interaction: It acts upon ”color” charge: red,
green, and blue. Color charge is conserved. Color charge is
carried by quarks, which come in 6 ”flavors, u, d, s, c, t, and
b. A proton is made of uud, and a neutron is made of ddu. It
is the strongest and short range (nuclear dimensions). It keeps
the nucleus together, so the positive protons will not fly apart.
It is mediated by the exchange of ”colored” gluons, which have
a color and anticolor, like red and antigreen.