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 . l
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 =
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 d
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
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
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
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
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 o
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 fr
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
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. Seco
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
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
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 (1
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 =