Winter 2000 Prof. A.M. Saperstein
Concept Summary (2/11/00)
(First half of semester)
1. Average speed is distance traveled divided by time of travel: d = vt
2.Average acceleration is the change in velocity divided by the time necessary
the change: (Delta)v = vfinal - vinitial
3.When starting from rest, traveling a time t at constant acceleration
accelerated motion), the distance traveled
= 1/2 a t2
4.For Aristotle, motion had to be explained; for Galileo (and successors),
it was change
of motion that required explanation.
5. For Aristotle, speed depended upon the ratio of propulsion to
6. For Galileo, in the absence of resistance, all bodies fall with the
same acceleration (which we call "g"
7. For Aristotle, vacuum was an impossible contradiction; for Galileo,
it was an ideal which could be approached.
8.Galilean inertia: an object moving on a level surface will continue to
move in the same direction at constant speed
9. Galilean superposition: motion in one dimension is independent of that
in a perpendicular dimension.
10. In the absence of resistance, the path of a projectile is a parabola.
11.Momentum is the product of mass and velocity. In a collision,
during any fully isolated physical process, momentum is conserved. Conservation
of momentum is equivalent to the inertia of the center-of-mass of a physical
system: if such a system is isolated, the velocity of its center-of-mass
12. "Velocity" implies both speed and direction: an object moving with
constant speed can still have changing direction and hence acceleration.
When going with speed v around a circle of radius r, there is a centripetal
acceleration: a = t2 / r.
13. Newtonian inertia - a generalization of Galilean inertia. In the absence
of forces, a body moves with constant velocity, i.e., with constant speed
(which could by zero!) in a fixed direction.
14. Newton's Force Law: When a body is accelerating (i.e., when 1
st law not applicable), the acceleration is proportional to the net
force acting on the body and inversely proportional to the mass of
the body. Force and acceleration are in the same direction F=ma.
15. An inertial frame is a reference frame in which Newton's first law
16. Galilean Relativity: all inertial frames are equivalent; no mechanical
experiment can tell you which inertial frame is moving, which not. Hence,
as far as the laws of mechanics go, there is no such thing as absolute
rest (or equivalent , absolute motion!)
17. Mass and weight are not the same! W = mg where g is the acceleration
due to gravity, the same for all bodies at a given location.
18. There is no experimental way of distinguishing between Ptolemy's model,
Tycho's model, and Kepler's model. It's all a question of "simplicity,"
"beauty," and context. Kepler's "laws" could be simply deduced from
Newton's Laws. The others could not.
19. Newton's Universal Law of Gravitation: For any two bodies, of
mass M1, andM2, located anywhere in the universe,
with a distance r between them, there is an attractive force, between them,
proportional to each mass and inversely proportional to the square of the
distance: F = G M1 M2 / r2 . It is
this gravitational force which is the weight of bodies at the earth's surface
20. Reductionists see a system as made up of parts where the properties
and behavior of each of the parts is essentially the same whether the part
is considered/observed separately or as part of the system. Holist's see
the parts of the system as essentially linked to the system as a whole.
The properties/behavior of each part would be fundamentally different if
it were to be removed from interacting with the rest of the system.
21. The kinetic energy of an object is proportional to its mass and the
square of its speed: K = 1/2 MV2.
22.Work is the product of the distance moved by an object and the
component of the force acting upon the object in the direction of its motion:
W = Fx.
23. It is necessary to do work on a body in order to change its kinetic
energy. Conversely, a body-while changing its kinetic energy - can do work
upon other bodies. Positive work implies an increase in kinetic energy
and vice-versa: W = DeltaK = Kfinal - Kinitial.
24.Work done on a body can change its "configuration" rather than its kinetic
energy, e.g., can compress a spring, stretch a rubber band, raise a body
against the pull of gravity. In this case, the work equals the increase
in potential energy. A decrease in potential energy means the body does
work on something else, e.g., a failing piledriver against a stake. Example:
work in raising a mass m through a height h at the earth's surface: W =
mgh (g the earth's gravitational acceleration: g = 9.8m/s2).
25.Or, a change in kinetic energy can produce a change in potential energy:
a ball thrown upward rises (increasing potential energy mgh) as it slows
(decreasing kinetic energy 1/2 MV2); Delta(1/2 MV2)=
- Delta(mgh). Thus in the absence of friction; the sum of K.E. and P.E.
is a constant. Mechanical energy is conserved! (M.E.= K.E. + P.E.).
26. In presence of friction; M.E. is decreased (not conserved), but heat
is generated. When methods for measuring heat (calories) were developed,
it was found that there is a unique relation between M.E. lost (friction)
and heat evolved and, conversely, between heat lost (heat engine) and M.E.
produced. Hence the conservation of energy law was extended to include
M.E., chemical , light, nuclear,....
27.In a closed system, energy can be neither created nor destroyed. Conversion
among the different forms of energy is allowed (with some restrictions-2nd
28.The K.E. of random motion of atoms and molecules of a substance manifests
itself as heat. The Temperature of the substance is a direct measure of
this random molecular energy.
29. When two objects are bound to each other, i.e., cannot freely escape
to infinite distance from each other, their total energy (sum of K.E. +
M.E.) is less than the minimum energy these two bodies would have if they
were free of each other. The difference in energies is called the "binding
energy." For example, for a space ship to escape the earth it must
be given sufficient kinetic energy ("escape velocity") to overcome
its binding energy to the earth.
30. Electrical charges are subject to a force law very similar to that
of gravitation though both repulsion and attraction are allowed. Under
"ordinary circumstances" electrical forces are much stronger that gravitational
31. Moving electrical changes constitute an electric current. Electric
currents produce magnetic forces upon other electric currents.
32. Gravitational, electrical, and magnetic forces between stationary masses,
electric charges, and electric currents, can be viewed as a "one-step"
33. Gravitational, electrical and magnetic forces can also be viewed as
a "two-step" process: masses create gravitational fields
in their vicinity; electric charges create electric fields in their
vicinity; moving electric changes create magnetic fields in their
vicinity. Then: gravitational fields at a point in space exert gravitational
masses placed at that point; similarly, electric fields exert electric
forces upon electric changes and magnetic fields exert magnetic forces
upon moving electric charges.
34. When masses, electric charges, or electric currents are moving or changing
with time, the "one-step" process (of #32) violates conservation of
energy and momentum. If we wish to keep these conservation laws, we
must believe in fields which carry energy and momentum. Interactions
are "two-step" processes and the fields are real.
35. Unlike particles, waves can show destructive as well as constructive
interference, i.e., the presence of two or more wave sources can result
in less wave energy at a point then that due to one of the single sources.
36. Waves travel with a velocity vwave which depends upon the
medium through which they are propagating: periodic waves are characterized
by a frequency f and a wavelength lambda,where f*lambda = vwave
37. Sound is a longitudinal wave through matter whose speed, in air, is
340m/s. Electromagnetic waves are transverse waves which travel through
a vacuum at a speed of 300,000 km/s.
38. Two periodic waves of the same type and frequency, coming from the
same source, will interfere constructively at a point when
the difference in distance they have traveled from that source equals a
whole number of wavelengths. Interference will be destructive
when this distance difference equals an odd integer number of half-wave-lengths.
39. Observers moving through a medium would expect (using our "common sence",
Galilean notions of space and time) to observe different wave speeds in
that medium than would be observed by observers stationary in that medium.
This is very commonly noticed with sound and water waves.
40. Light is believed to be a wave (electromagnetic wave
- Maxwell) since it demonstrates wave interference; yet no difference in
wave speeds due to motion has ever been observed (Michelson and Morley).
Thus either a medium for light propagation (aether), in the usual sense,
does not exist, Galileo-Newtonian notions of absolute space and time are
wrong, or both.
41.Lorentz and FitzGerald gave an ad-hoc explanation of the failure of
Michelson and Morley to detect motion with respect to the electromagnetic
medium by saying that matter physically shrank in the direction of motion
since all matter was held together by electrical forces.
42. Einstein wished to preserve the validity of Galileo relativity and
of Maxwell's equations. Hence he postulated that nature was
such that the velocity of electromagnetic waves in a vacuum was the
same for all observers. From this postulate, using simple, straightforward
logic, followed profound changes in our concepts of space and time.
43. Simultanaity is no longer absolute. (No longer a universal absolute
time.) Different observers, can disagree to the time ordering of non-co-located
44. Observers moving with respect to each other will disagree about the
length of an object in the direction parallel to their relative motion.
45.Not only are there differences in the "minds-eye" construction of reality
done by different observers - due to their agreement on the speed of light
and consequent disagreements on space and time intervals, there are also
differences as to what they actually see due to the finiteness of the speed
of light. (The distortions seen by
different observers would be there even if Galilean notions of
space and time were correct, provided that the motions involved were fast
46. Let mo be the mass of an object at rest with respect to
observer 0 as measured by O; let Lo be a length of the object,
as measured by 0, and To be a time interval associated with
the object, again as measured by 0. Let the observer 0' be moving with
velocity v with respect to 0, in a direction parallel to the length
Lo, and let m, L, and
T be the corresponding quantities as measured by 0'. Then,
1/gamma = Square Root [1 - (v/c)2]. Lo, To,
and mo are called the "proper" or "rest" mass,
length, and time of the object. Note that gamma is always greater or equal
to one so that mass and time are always greater than or equal to rest mass
and proper time, while length is always less than or equal to rest length.
47. Mass is a form of energy (or energy is a form of mass) via the
relation E= mc2
48. (Delta S) 2 = (Delta x) 2 + (Delta y) 2
+ (Delta z) 2 - c 2 (Delta t)2 is a relativistic
invariant (the same
for all inertial observers) interval between two events occurring
at the two space-time points (x1, y1, z1,
t1 ) and (x2, y2, z2, t2)
where Delta x = x1 - x2, Delta y = y1
- y2, Delta z = z1 - z2, and Delta t =
t1 -t2. If (Delta S) 2 is positive,
the interval is "space-like". There is a reference frame for which
the two events occur at the same time (t1 = t2) with
a proper distance Lo = Delta S between them. If (Delta S) 2
is negative [(Delta S) 2 = - (Delta t)2],
the interval between the two events is "time-like"; there is a reference
frame for which the two events occur at the same location [x1
= x2, y1 = y2, z1 = z2,
but at different times, one in the future of the other.
49. All observers will agree on the past-future relationship between two
events which are separated by a time-like interval (i.e., past and future
are absolute for such a pair of events). Observers may disagree as to past
or future for events separated by a spacelike interval (i.e., for these
two events, past and future are "relative".)
50. The principle of equivalence states that it is impossible to experimentally
distinguish between a gravitational field and an accelerated reference