Making Graphs!

X(t) = XmaxSin(ωt + π/2)

Adjusting the Period
Fs = -Kx
F = ma
ma = -Kx, a = -k/m * x

K = 50 n/m
T = 1sec
m = 1.27Kg

The simple pendulum
SHM: a = -?x
F = ma
-mg*sinΘ = ma

Gmean = 1.6m/s^2
ge = 9.8 m/s^2
Te = Tm
Lm = lm/le Le
= 1.6/9.8 Le

Where am I?!

1) Latitude
2) Longitude

Latitude is easy to exactly find out.
Calculating latitude, tropic of cancer and equator
α is exact same degree of sun shadow

Longitude is completely arbitrary

How to keep the time?
-Need Periodicity

We need chronometer
-periodic, isochronous, predictable

Equilibrium
c = 2πr
2πradians = 360°

cos(3π/4)=-0.707
Sin(3π/4)=0.707

Angular Velocity w
ΔΘ=ωt

A Humbling Moment

All objects are mutually attracted
– Accelerate at 10m/s^2

The Train… of Pysics!
What trajectory would the apple follow?

In physics motion at constant speed is indistinguishable from being at rest!

F:10N
time:4.47sec
acceleration:1 m/s^2

Δx=v0t+1/2at^2
10m = 0 + 1/2a(4.47)^2

N = Normal(perpendicular)
FN mg=weight, mg is equal to FN

Forces and trig
if α=30°、Fr=100N sin(30degree)*100
FRX = 50N, Fry = 86.6N cos(30degree)*100

m = 10Kg
mg = 100N
Fry = 50N
Fn = 150N
Frx = 86.6N
Fw = 86.6N

Components of Motion

2-D motion can be solved as two
1-D motion problems

Δx = V0t + 1/2αt^2
V = V0 + at
V^2 = V0^2 + 2αΔx

Δox = 10m/s
α- 10m/s^2

Dimensional Analysis
E = mc^2, E = M*L*T

Experiment, Evidence Rules, Motion Equation
1D/ 2D ΔX = v0t + 1/2at^2

Explain the cause of motion
Understand “rest”

the motions of stars and apples have in different, but governed by same laws

Why learn about motion?

Kinematics
1. predictive power
2. important tool
3. solve nature’s puzzle

Galileo Galilei:object fall at constant speed
Aristotle vs Galileo: objects gain speed as they fall

A mass dropped from greater height leaves greater impact
Feathers do fall at constant speed
Air slows objects down! … too fast

Position, Velocity
Velocity: v = Δx / Δt
Acceleration a = ΔV / Δt

v = 5m / 1s = 5 m/s

Average deviation
2.00m, 1.20m, 2.80m -> 1.6/3 => 0.53
1.97m, 1.98m, 1.97m -> 0.02/3 => 0.0067
m / s / s
“constant acceleration means an object gains equal speed during equal time intervals”
Vαt^2
x = t^2, a = 10m/s^2 = g
V0 = 10m/s

latitude

latitude
https://www.esrl.noaa.gov/gmd/grad/solcalc/

Shadows and Trigonometry
18m, 10m, β
β = arctan(18/10) in degrees = 60.9°

w = 25m / tan(1.3°) = 1101m

MC = 40,000Km / 3.68, MC = 10,900K

distance to the moon
10900 = πd
d = 3470km
tan(0.5°)=3470 / L
L = 3470 / tan(0.5) = 398,000 km

distance to the sun
cos α = 400,000/L
L = 400,000/ cos89.853°
L = 156,000,000 km

trigonometric ratios

tan, sin, cos
e.g. tanα = 231 /266

using angle table
534.6m, 401.0m

length of bar = 1.00m, length of shadow = 0.126m, α = 7.2°

Eratosthenes vs Reality
try out in Google
cos(44 degrees) = 0.7193319

reverse tangent -> arctan(0.932) in degrees -> 42.98
sin(30 degree) = 0.5
tan(79 degree) = 5.14
arcsin(0.62) in degrees = 38.32°
arctan(4/7) in degrees = 29.74°
arctan(1/40) in degrees = 1.42°

α = 7.2°+- 1.4°
αmin = 5.8°, C = 57,400km
αmax = 8.6°, C = 46,250km

Landmark of physics

Eratostheness

How long the earth’s circumstance?
Geometry sun’s position

Plato “The earth’s circumference is 400,000 stadia.”
≒ 74,000
1 stadium = 185m

True Circumference is 40,000 Km
Plato’s guess was 85% larger than the true circumference.

Archimedes: circumference of Earth is 300,000 stadia
Archimedes: Look at the geometry of a sphere and find a clever way to calculate the circumference