Thermodynamics and statistics

Power

Temperature

Heat energy

Distributions

Clicker Question #6

When I toss a ball, while it’s moving up:

a) Its kinetic energy increases, while its potential energy decreases

b) Its kinetic energy decreases, while its potential energy increases

c) Its kinetic energy increases, and its potential energy increases

d) Its kinetic energy decreases, and its potential energy decreases

e) Its kinetic energy and its potential energy remain unchanged

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Power=rate of transfer of energy

• Power is simply energy exchanged per unit time, or how fast you get work done: P = DE / Dt

• watts (W) = joules/sec

• One horsepower = 745 W

• Perform 100 J of work in 1 s, and call it 100 W

– 100W light bulb converts electrical into radiant and thermal energy

• Run upstairs, raising your 70 kg (700 N) body 3 m (=2,100 J) in 2 seconds 1050 W avg energy expenditure

• Food energy goes into GE (+ heat)

• Space Shuttle puts out a few GW (gigawatts, or 109 W) of power!

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Thermal physics: temperature and heat

• As with motion: start by defining terms to describe phenomena

• Heat = energy in the form of random molecular motion

– KE of molecules within body

– Contrast with: collective motion of molecules = KE of a moving body

• Temperature

– Everyday definition: "what you measure with a thermometer"

– Deeper definition: measure of average KE of molecules in "system"

• system = solid body, liquid or gas in a container, Universe, etc

– Units for temperature

• Degree Celsius (US usage: "Centigrade")

0°C = freezing point of water, 100°C = boiling point of water

• Fahrenheit (US only): 0°F = coldest day, 100°F = hottest day

– recall from high school: T(°C)=(5/9)(T(°F)-32)

• kelvins (K): "absolute" temperature scale

Same size as Celsius degree

0°K = "absolute zero" (-273C), so room temp ~ 293K

(Lord Kelvin, 19th C.) We now say "kelvins", not "degrees Kelvin"

Temperature for "zero" molecular KE (but it‟s never quite zero!)

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Thermal energy

• Conservation of energy is also known as the

First Law of Thermodynamics

• Thermal energy (“heat”) is another form of energy – just randomized kinetic energy on micro scale

– atomic vibrations in solids, motion of molecules in liquids/gases

• Energy goes into heat in many ways – Product of friction, and many chemical or electrical processes

• Hard to make heat energy do anything for you – Kinetic energy of hammer can drive nail

– Potential energy in compressed spring can produce motion

– Water behind dam can fall to drive generators

– Heat is usually too disordered to herd into useful work

• notable exceptions: steam turbines found in power plants – just one type of heat engine

• Sun: heat is important in enabling thermo-nuclear fusion

“motion of heat”

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How well do we know things? Uncertainty in measurements

• Afflicts all measurements

• When scientists talk about the “errors” associated with a measurement, they really mean uncertainty

– Not a “mistake”, just a natural part of the measurement process

An Example:

• What time is it now?

– If you have a watch, you “know” what time it is…

– But if you have 2 watches and they disagree, what time is it?

– If you and your fellow students have 211 watches, and they all are slightly different – how do you decide what time it “really” is?

• You might choose to take an average

– You might choose to re-average after throwing out obviously wrong watches!

• You might choose to use only the most expensive watch among the 211, and declare it to be „true time‟...

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Systematic vs. Random Uncertainty

• “Errors” fall into 2 categories

– Systematic uncertainty = bias or offset in the measurement

• Always the same under same circumstances: may be correctable

– Random error = fluctuation in reading which could go either + or –

• Can only be estimated statistically

Why wasn‟t there a single answer to “what time is it”?

• Examples of systematic uncertainties

– Calibration

• Was the clock set correctly initially

• Does it tick at the proper rate (ie 1.0 sec per tick)

– Did we forget to adjust for daylight savings?

• Examples of random errors

– Measurement errors: interpreting where the clock‟s hands are

• Different observers can report different times from the same clock

• Same observer might get different errors on different occasions

– Digital data makes this harder to mess up !

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Statistical errors

• Every experimental measurement has some level of (in)accuracy:

– Systematic error: applies to all data

• Systematic errors can often be corrected

– Random error: each measurement is subject to random fluctuations

• Example: you measure identical time intervals with a stopwatch, and get a slightly different time each measurement, due to your reflexes

– For measurements that involve counting randomly-occurring events (like cosmic ray detections), "it turns out"* that the random error s = N

• Example: run cosmic ray detector 1 day, collect 9 "events"

– We'd say the number of events per day is 9 + 3

– We show the estimated error by error bars on the data point:

This means: we measured 9, but the true value has a probability of 68% of being anywhere within the "error bars"

* Code for "too hard to explain now!" - but we'll see why 68% soon

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sigma

square root

9 9

length of paces

0

1

2

3

4

5

6

7

8

9

10

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

1.1

1.2

meters

Nu

mb

er

of

stu

de

nts

`

mean

standard deviation

(central 68% of sample)

A special kind of graph: histogram

• When we discuss statistics, it is useful to make "histograms" showing how often something occurs

– Bar graph showing (# of occurrences) vs (whatever you‟re measuring)

• Example: pace (step) length of PHY 110 students

Here are some statistics for the pace measurements:

number of students 58

max. length, in m 1.23

min. length, m 0.35

mean, m 0.81

"standard deviation” 0.17m

Definition: statistic = a single number that characterizes a whole set of data

But a picture is simpler: Avg pace = 0.81 + 0.17 m

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Basic Statistics Jargon

• There are many commonly used “statistics” which summarize long lists of data into a handful of numbers that are simpler to digest

– Average (“mean” value): add up all x values, and divide by N

– Median: sort the data in order of increasing x, and take the middle x value

– Mode (peak, most probable value): x value for the highest point on histogram

These statistics all try to indicate the “center” of the histogram in a useful way

– Standard Deviation (s, Greek sigma): a measure of the width of the histogram

• Assumes the “underlying probability distribution” is “normal” (bell-shaped)

s is commonly used to show how tightly data cluster around the “central value”

0

0.01

0.02

0.03

0.04

0.05

-3 -2 -1 0 1 2 3

x

p(x

)

Mode Median

Mean

Mode, median and mean

Standard deviation, s

For a symmetrical data distribution, the 3 coincide

For a skewed data distribution, the 3 are not the same

This is the famous

“Bell-shaped curve”

probability distribution

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Bell-shaped curve shows up everywhere… Example: heights of Americans

• Men: 69 ± 3 inches (standard deviation is 3 inches), which means:

– 68% are between 5‟6” and 6‟0”

– 95% between 5‟3” and 6‟3”

– 99% between 5‟0” and 6‟6”

• only 1 in 700 men taller than 6‟6”

• Women: 65.5 ± 2.5 inches, meaning

– 68% between 5‟2” and 5‟7”

– 95% between 4‟11.5” and 5‟9.5”

– 99% between 4‟9” and 6‟0”

Heights of Americans

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

56 60 64 68 72 76

height, inches

Pro

bab

ilit

y

Men

Women

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Why is the “normal” distribution so important ?

• The Gaussian or “Normal” distribution describes a huge variety of phenomena – How far a bullet misses its target, in repeated shots

– Pattern of wear on stone steps in ancient monasteries

– Scores students get in a (fair) exam

Why?

• “Central limit theorem”: for large samples, – If you add up contributions from many random processes, the sum

will follow the normal distribution regardless of how each contribution may be distributed

• Most physical observations are actually a sum of many steps: – Particle leaves a track of some length

– Particle detector measures length of track

– Electronic device measures detector response

– Data recorder digitizes electronic signal

Each step introduces some uncertainty: final “data point” has error which amounts to sum over individual errors in steps

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Simple example: try this at home

• Pick 10 numbers, randomly between 0 and 9

– Easy source: use the last digit of telephone numbers in the white pages (yellow pages = businesses: some choose their numbers, so it‟s a biased list)

– Or, if you know how to use Excel, you can do this quickly using built-in functions

• Add up the 10 random numbers and write down their sum, S = N1 + N2 + N3 + … N10

• Repeat the whole process 10 or more times and use results to draw a histogram

– S values will begin to look like a normal distribution!

– You can see why

• Very small or very large S requires ALL 10 random numbers to be close to 0 or close to 9!

• It‟s much easier to get a middling S value

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Example: your 1st midterm scores

• Large enough

sample to see the

onset of normal

distribution

• But cutoffs at

both high and low

ends

• “Curve”: I calculate the “z-score”:

z-score = (your score – class average)/(standard deviation)

z-score = 0 if you have average; z-score > (<) 0 if above (below) mean

Mean = 16.48 St.dev. = 2.74

Clicker Question #7

The most probable value of a distribution is also known as

a) The mean

b) The average

c) The mode

d) The median

e) The standard deviation

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