General Formula for Error Propagation

( , , )( , , )best best best best

q q x y zq q x y z

==

for independent random errors δx, δy, and δz

22 2q q qq x y zx y z

δ δ δ δ⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

main formula for error propagationalways use this formula

Experiment 1: Measure Density of EarthExperiment 1: Measure Density of EarthExperiment 1: Measure Density of Earth

• Calculate average density ρ and determine which elements constitute the major portion of the earth.

• Two measurements– (a) Earth’s Radius Re . (challenging measurement) – (b) Local acceleration of gravity g. (fairly easy)

• Use Newton’s constant G=6.67 X 10-11 N m2/kg2

• Aim for 10% or better error on ρ.

Gravitational force2

GMmFr

=34

3 432 2

( )ee

e e

F GM G Rg GRm R R

π ρ π ρ= = = =

34 e

gGR

ρπ

=

What’s the PointWhatWhat’’s the Points the Point

Its an experiment about optimizing measurement technique, error estimation, and error propagation

What Element(s) make up the EarthWhat Element(s) make up the EarthWhat Element(s) make up the Earth

• Assume most of earth’s volume is one element.

rock 2.7aluminum 2.7zinc 7.14iron 7.20nickel 8.85copper 8.89silver 10.5lead 11.34mercury 13.60gold 19.3

Densities

Measure Earth’s Radius using ∆t SunsetMeasure EarthMeasure Earth’’s Radius using s Radius using ∆∆t Sunsett Sunset

e

e

Re e

R 1cos( ) 1R 1 Rh

hh

θ = = ≈ −+ +

From right triangle:

2

cos( ) 12

θθ ≈ −

For small θ :

Equating:2

e2 Rhθ

=

e

2R

hθ =

Assume we are at equatoreRh

Sunset at height h

Sunset on earth’s surface (tangent)

eReR h+θ

View from North Pole

Apparent motion of the sun as it sets.

θ Increases at Earth Rotatesθθ Increases at Earth RotatesIncreases at Earth Rotates

Earth makes (nearly) one rotation per day.Angular frequency is 2π radians per day.

5

radians2 radiansday 7.27 10 secondhours minutes seconds24 60 60

day hour minute

πω −= = ×

⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

e

1 2R

htω

= Solving for t, we get the time delay of the sunset at height h (since the true sunset).

ω (omega) = earth’s angular frequency.

e

2R

htθ ω= = θ (theta) = angle earth rotates after true sunset.

S

λλs

winter view

Correct for Latitude and Earth’s AxisCorrect for Latitude and EarthCorrect for Latitude and Earth’’s Axiss Axis

32.8723.4 sin(2 / 365)s d

λ

λ π

=

= −

o

o

La Jolla latitude

Solar latitude varies.d=days since Sept. 22

(or March 20).

2 2 2 2e e

1 2 1 2R [cos ( )cos ( ) sin ( )sin ( )] Rs s

h Chtω λ λ λ λ ω

= ≡−

This formula accounts for our latitude and for the This formula accounts for our latitude and for the angle of the earthangle of the earth’’s axis from the plane of its orbit.s axis from the plane of its orbit.

Measuring the Height of the CliffMeasuring the Height of the CliffMeasuring the Height of the Cliff

h1l

h=l cos(Θ)Θ

• The formula we derived is for height above sea level.

• Strings, protractors, and rulers will be available.

• Be sure to understand how well heights must be measured before you do the experiment.

• Each pair of experimenters should get their own measurements.

Your Height Above Sea Level on BeachYour Height Above Sea Level on BeachYour Height Above Sea Level on Beach

h2

• The experimenter on the beach also views the sunset from above sea level.

• When you check the error propagation you will find that the measurement of the earth’s radius is quite sensitive to the h2 measurement.

“The Equation” for Experiment 1a““The EquationThe Equation”” for Experiment 1afor Experiment 1a

2

1 2e 2

2Rh hC

tω⎛ ⎞−

= ⎜ ⎟⎜ ⎟∆⎝ ⎠

use this formula for your error analysis

From previous page.e

1 2RCht

ω=

Time difference between the two sunset observers.( )1 2 1 2e

1 2RCt t t h h

ω∆ = − = −

Season dependant factor slightly greater than 1.2 2 2 2

1cos ( )cos ( ) sin ( )sin ( )s s

Cλ λ λ λ

≡−

Propagating Errors for RePropagating Errors for Propagating Errors for RRee2

1 2e 2

2Rh hC

tω⎛ ⎞−

= ⎜ ⎟⎜ ⎟∆⎝ ⎠basic formula

1 21 2

e

e e eR t h h

R R Rt h h

σ σ σ σ∆∂ ∂ ∂

= ⊕ ⊕∂∆ ∂ ∂

Propagate errors (use shorthand for addition in quadrature)

( ) ( )1 21 1 2 2 1 22

e

e e eR t h h

R R Rt h h h h h h

σ σ σ σ∆= ⊕ ⊕∆ − −

Note that error blows up at h1=h2 and at h2=0.

Cliffs West of Muir CampusCliffs West of Muir CampusCliffs West of Muir CampusAt the bottom of the asphalt road is a reasonable place to measure.

Must return there at sunset.

Do not go too near the cliffs

Do not drop or kick objects below on the beach

Wear walking shoes

It may be cold in the evening

Weather plays a role.Completely clear days are best.

sunset time – a moment when the last point of the Sun disappears

Measuring g with a PendulumMeasuring Measuring gg with a Pendulumwith a Pendulum

• Period can be measured with electronic timer over one cycle of with a stopwatch over many cycles.

• Frictional forces play a role for light weights.

• Small oscillations are good.• Heavy weights may cause coupling to

other oscillators like unstable stand.• Short strings may cause moment of

inertia to become important.

Period of pendulum2 lTg

π=

Propagating Errors for Experiment 1Propagating Errors for Experiment 1Propagating Errors for Experiment 1

34 e

gGR

ρπ

= Formula for density.

2

3 1 34 4 eg Re e

gGR GRρ

σ σ σπ π

−= ⊕ Take partial

derivatives and add errors in quadrature

eRg

eg Rρ σσ σ

ρ= ⊕

Statistical analysis

uncertainty

Period of pendulum2lTg

π=

error propagation statistical analysis two methods to find uncertainty

The mean

the best estimate for x → the average or mean

sigma notation

N measurements of the quantity x

common abbreviations

1 2

1 2

1 21

, , ... ,

...

...

N

best

iN

N

i i i Ni i

x x xx x

xx x xxN N

x x x x x x=

=

+ + += =

= = = + + +

∑

∑ ∑ ∑

The standard deviation

deviation of xi from x_

standard deviation of x

RMS (route mean square) deviation

average uncertainty of the measurements x1, …, xN

uncertainty in any one measurement of x → δx = σx

68% of measurements will fall in the range xtrue ± σxxtrue + σxxtrue – σx

xtrue

x

i id x x= −

( )21x ix xNσ = −∑

( )211x i

x xN

σ = −− ∑

The standard deviation of the mean

uncertainty in xis the standard deviation of the mean

_

based on the N measured values x1, …, xN we can state our final answer for the value of x:

(value of x) = xbest ± δx

xx

x

Nx

σσ

δ σ

=

=

bestx x= xxx Nσδ σ= = ( )value of xx x σ= ±

ExampleWe make measurements of the period of a pendulum 3 times and find the results: T = 2.0, 2.1, and 2.2 s.(a) What is the mean period? (b) What is the RMS error (the standard deviation) in the period?(c) What is the error in the mean period (the standard deviation of the mean)?(d) What is the best estimate for the period and the uncertainty in the best estimate.

2 2 2 2 2 2

1 1 (2 2.1 2.2) 2.1 s3

1 1 1( ) (2 2.1) (2.1 2.1) (2.2 2.1) 0.1 0.1 0.1 s1 2 20.1 0.057735 s 0.06 s

32.10 0.06 s

i

T i

TT

T

T TN

T TN

NT T

σ

σσ

σ

= = + + =

⎡ ⎤ ⎡ ⎤= − = − + − + − = + =⎣ ⎦ ⎣ ⎦−

= = = →

= ± = ±

∑ ∑

∑

( )211x i

x xN

σ = −− ∑

xx N

σσ =1 ix xN= ∑ ( )value of xx x σ= ±

Systematic errors

random component systematic component

( ) ( )22ran sysx x xδ δ δ= +

1 2 3 4 5 6 7 8 9 10