Introduction
Jack and Jill went up the hill To fetch a pail of water. Jack fell down and broke his crown and Jill came tumbling after.
Every physical quantity has units . laws of people dimensions . laws of nature
Units English (traditional) French ("scie
Harmonic Motion
Spring dx k = x 2 dt m Solution x = A cos t + B sin t k = m 2 T= 1 f= T
2 2
Pendulum d m 2 ( L ) = mg sin dt 2 d g = 2 dt L
2
=
g L
Why did m not cancel for the spring?
kx0 mg = 0
F = k ( x0 x ) mg = kx
dx m 2 = kx dt
2
Energy 121 212 kB =
Gyroscopes
Bicycle wheel on a rope u r r r u dL r = rF = into the plane dt u r u r Spin the bicycle wheel L = I c u r r dL = into the plane dt
Rate of precession d = dt u r r dL = dt
r r dr r v= = r dt d dt u r r dL = = L dt mga = ( I c ) mga = I c
Lz = c
Translation and Rotation
Kepler's Three Laws 1. Law of ellipses 2. Equal areas in equal times 3. T a
3 2
Modern argument uuu rrr A B = C C = area of parallelogram u r 1r r r r = A 2 divide by t 0 u r r d A 1 r dr 1 r r = r = r v dt 2 dt 2
u r r r dA 1r u
Rotation of Rigid Bodies
u 1r r R= rdm M or better, 1r 0= rdm M
Displacement
Rotational
x
dx v= dt
d = dt
d = dt
dv a= dt
r r = vector from origin l = distance to axis r =x +y +z
2 2 2 2
l =x +y rur r v =r
2 2
2
v = l
Kinetic energy 1 2 K = mi vi i2 vi =
Momentum
Single body u r u dp r F= dt u r u r If F = 0, p = constant
Two bodies u r u r F 12 = F 21 d ( m1v1 + m2v2 ) = 0 dt u r Total p = constant provided no external forces
Any number of bodies u r u r u r F i , j = L + F k ,l + L + F l , k + L = 0
i,
Out of Gas
Out of Gas
The end of the age of oil
David Goodstein
Energy Myths
$3 a gallon is too much to pay Oil companies produce oil. We must conserve energy. When we run out of oil, the marketplace will take over Theres enough fossil fuel in the ground
Conservation of Energy
h FP = mg sin = mg s h a=g s v = at 12 s = at 2 2s t = a
1 2
2s v = a = 2 sa a v = 2 gh
1 2
Same argument runs in reverse
Law of Conservation of Energy Kinetic energy: energy of motion v Potential energy: energy of position h
2
KE
Non-inertial Frames
r s ( t ) = vx ti u r r dr ' dr = vx i dt dt u r r 2 2 r ur u d r' d r = 2 or, a = a ' 2 dt dt u r r ur u F = ma = ma '
Any frame in which F = ma is an inertial frame. Any other frame moving at constant speed in a straight line with re
Forces of Nature
uu r m1m2 Fg = G 2 r r uu r q1q2 Fe = K e 2 r r
mM e F = ma = G 2 Re GM e a=g= 2 Re
G = 6.7 10
9
11
Nm / kg
2 2
2
2
K e = 9 10 Nm / C
1. Spring d 2x m 2 = kx dt 2. Viscosity Fv = 6 Rv 3. Friction F fr = N
m1m2 Fg = G 2 r r q1q2 Fe = K e
Shoot the Monkey
Newton's Laws First Law: Every body continues in its state of rest or of uniform motion in a straight line unless it's compelled to change that state by forces impressed upon it. Second Law: The change of motion of an object is proportion
Trajectories
r r r lim r ( t + t ) r ( t ) d r r = = v( t) t 0 t dt r d r dx =i + dt dt dy dz +k j dt dt
r r = i ( r cos ) + ( r sin ) j r v = i ( r sin ) + ( r cos ) j r ( 2 r cos ) ( 2 r sin ) a = i j r 2 = r v = r r
2
t ]
Uniform circular motion r = co
Vectors
uuu rrr 1. A + B = C uuuu rrrr 2. A + B = B + A uuu rrr 3. A B = C rr u r r 4. s = vt and F = ma
u uu uu uu r rrr 5. A = Ax + Ay + Az + Ay + Az k = Ax i j
uuu rrr A+ B = C
(
Ax i + Ay + Az k + Bx i + . = ( Ax + Bx ) i + . j
)(
)
Cx = Ax + Bx C y
The Law of Falling Bodies and The Calculus
The Law of Falling Bodies In a vacuum all bodies fall with the same constant acceleration
1. Aristotle: natural place. Heavier=faster. 2. Albert of Saxony (12th C): v s 3. Nicole Oresme (14th C): v t 4. Nicole Or
Resonance
dx m 2 = kx dt
2
dx m 2 = kx dt x = A cos t + B sin t
2
dt x = A cos t + B sin t
m
dx
2
2
= kx
=
2
k
T=
m 2
Fd = bv dx = b dt
dx dx m 2 = kx b dt dt
2
x = Ce cos t Transient solution
t
x = A sin t + B cos t
[
[ L] sin t + [ L] cos t = 0
=
2 0