Transcript Lecture 1

What is PHYSICS?
study of matter and energy (physical world)
delusionary attempt to find order in dirt and cosmos
quintessential reductionist paradigm (= most basic science)
different kinds of sciences
(different from engineering in objectives)
Need some numbers to work on.
Let’s go measure something!
What is to be measured?
How to measure it?
Problems & limitations
Ways to circumvent
Measurement standards
Many units to match the dimension of an item
or quantity being measured.
Time: second, minute, hour, day, year, century,
millenium
Length: centimeter, meter, kilometer
inch, foot, yard, rod, mile
Matter: gram, kilogram, metric ton
ounce, pound, slug
Force (derived unit): Newton, pound (lb)
How well can these measurements be made?
In principle, any arbitrary precision.
In practice, limited by instrument and method.
Express precision by significant figures and
scientific notation.
Look at the statistical basis of measurement data.
Measure the diameter
13.7 cm
Scientific Notation
useful for expressing large dynamic range
(keeps track of the decimal point)
and significant figures
m.mmm  10eee
( m.mmmEeee on a calculator )
where
1.0  m.mmm < 10.0 and eee can be + or 
• Workshop Physics is a new method of
teaching introductory physics without
formal lectures.
• Instead students learn collaboratively
through activities and observations.
Observations are enhanced with computer
tools for the collection, graphical display,
analysis and modeling of real data.
• Typical Workshop Physics classes meet for
two 3.5-hour long sessions each week and
students use an Activity Guide.
• In developing Workshop Physics it is
assumed that the acquisition of transferable
skills of scientific inquiry are more
important than either problem solving or
the comprehensive transmission of
descriptive knowledge about the enterprise
of physics.
• There were two major reasons for the
emphasis on inquiry skills based on real
experience.
• First, the majority of students enrolled in
introductory physics at both the high school
and college level do not have sufficient
concrete experience with everyday
phenomena to comprehend the
mathematical representations of them
traditionally presented in these courses.
• The processes of observing phenomena,
analyzing data, and developing verbal and
mathematical models to explain
observations, afford students an opportunity
to relate concrete experience to scientific
explanation.
• A second equally important reason for
emphasizing the development of
transferable skills is that, when confronted
with the task of acquiring an overwhelming
body of knowledge, the only viable strategy
is to learn some things thoroughly and
acquire methods for independent
investigation to be implemented as needed.
• Although lectures and demonstrations are
useful alternatives to reading for
transmitting information and teaching
specific skills, they are unproved as vehicles
for helping students learn how to think,
conduct scientific inquiry, or acquire real
experience with natural phenomena.
• The time now spent by students passively
listening to lectures is better spent in direct
inquiry and discussion with peers.
• Many educators believe that peers are often
more helpful than instructors in facilitating
original thinking and problem solving on
the part of students.
Statistical Measures
• Systematic errors: consistent influence on
measurements which can increase or
decrease all values in the same direction.
Examples?
• Ruler too long or short, or bent.
• Uncertainty is a fact of measurement.
• How do you know if systematic error is
present?
• Random errors: inconsistent influence on
individual measurements which can usually
be eliminated. Why can we say this?
• If we perform the measurement a
significant number of times, the high and
the low uncertainties will cancel out each
other. The bell curve.
• Examples?
• Statistics deals with random errors
Weigh some breakfast cereals
Raisin Bran (g)
Tasteeos (g)
Honey Nuts (g)
65.6
40.8
43.6
63.1
42.9
45.1
67.7
42.3
44.2
66.0
41.9
45.8
66.4
42.5
44.8
67.7
40.5
45.1
Consider the average Tasteeo
1
2
3
B
Tasteeos (g)
40.8
42.9
4
5
6
42.3
41.9
42.5
7
8
9
40.5
41.816666667
0.964192235
Under Excel, highlight the B8
cell and insert the AVERAGE
function
AVERAGE(B2:B7)
Next, highlight the B9 cell and
insert the STDEV function
STDEV(B2:B7)
Consider the average Tasteeo
L1
40.8
42.9
42.3
41.9
42.5
40.5
L2
L3
With the TI-83, enter the
column of data using the
STAT editor.
Return to STAT, select
1-Var Stats.
This will be returned to
the main screen, so now
1-Var Stats L1
And obtain:
•1-Var Stats
Where is the average?
•x = 41.81666667
•x=250.9
•x2=10496.45
•Sx=.9641922353
•σx=.8801830618
•n=6
Where is the std dev?
Beware of σx (what is it?)
Are all digits significant?
Mathematical Preliminaries
• Data is often repeated measurements of the
same quantity.
• A “reliable” central measure of the data is
the mean (average).
• The first moment of the distribution, the
standard deviation, is related to the
probability that each measurement is close
to the mean.
• Standard deviation tells us how close an
additional measurement would come to the
center distribution of an infinite number of
measurements.
• We assume that our finite average comes
close to the infinite mean.
68% of the measurements fall within 1 standard
deviation from the mean
95 % would fall within 2 STDEV’S from the mean
• Some data appears to form a normal
(gaussian) distribution on a histogram.
Even if it doesn’t, it is convenient to model
the data as gaussian to calculate the Std
Dev.
• Another reliable measure for the data is the
standard deviation of the mean (SDM).
This expresses the probability that the mean
can vary. The SDM is gaussian for large
sample sizes.
• There are higher moments of the distribution
which are informative in some situations
(e.g., skew, kurtosis).
Behold the Histogram!
Fig. C.1. A histogram representing the variation in a set of measuremen
proportional to the number of measurements in each small range of values.
A histogram representing the variation in a set of
measurements. The height of each bar is proportional to
the number of measurements in each small range of values.
Consider the definition of the mean and the
std dev (standard deviation).
Let each of the N measurements be called xi (where i = 1 to N) and let the average of the N values
of xi be . Then each residual ri = xi – . Thus:
=
 x1  x2  x3  x4  x N  
(C.1)
N
ri = – xi
(r1  r2  r3  r4 rN )
( N  1)
2
SD = sd =
=
2
2
r
2
2
2
i
( N  1)
Here the symbol means “sum the terms i = 1 to i = N.”
A test of significance is if any new data is beyond
the 95% (“2σ” or two standard deviations) level.
. A Gaussian distribution curve showing the 95% confidence interval.
A smooth Gaussian distribution curve
showing the 95% confidence interval.
• Generally one arrives at a best estimate
of a measurement of interest by making a
series of measurements and averaging the
results.
The standard deviation is a measure of the
level of uncertainty in the data.
Standard Deviation of the Mean
SDM =
 sd
N
It is this quantity that answers the question, “If I repeat the entire series
of N measurements and get a second average, how close can I expect this
second average to come to the first one”? Standard Deviation of the
Mean is SDM: sometimes the SDM is referred to as standard error. Since
the SDM is actually a measure of uncertainty rather than of an error, in
the sense of a mistake, we prefer not to use the term standard error.
Sample Calculation of Standard Deviation and SDM
Let us imagine that you made the following series of length measurements
with a good centimeter ruler: 12.2, 12.2, 12.3, 12.0, 12.1 cm.
1.
2.
3.
4.
5.
Measure
12.2
12.2
12.3
12.0
12.1
Sum: 60.8
Mean
-12.16=
-12.16=
-12.16=
-12.16=
-12.16=
Residual
+0.04
+0.04
+0.04
+0.16
-0.06
Residual2
0.0016
0.0016
0.0196
0.0256
0.0036
Sum:
0.0520
Average:
60.8 /5 = 12.16 cm
Sum of residuals squared:
0.0520
Sum of residuals squared divided by (N–1):
0.0520/4 = 0.0130
Standard deviation:
sd =
0.0130  0114
.
cm
Standard deviation of the mean:
.
SDM = 0114
5  0.051 cm
Reported result:
L = 12.16 ± 0.05 (SDM) cm
Estimating Volume
Vol =  r 2 h
 3  (5.0E2 m) 2  (10 m)
 8E6 m3 (8 106 m3 )
Estimating Speed
According to a rule-of-thumb, every five seconds between
a lightning flash and the following thunder gives the distance
of the storm in miles. Assuming that the flash of light arrives
in essentially no time at all, estimate the speed of sound in m/s
from this rule.
1 mi 1609 m

 322 m/s
5s
1 mi
Kinematics in One Dimension
MECHANICS comes in two parts:
kinematics: motion (displacement, time, velocity)
x, t, v, a
dynamics: motion and forces
x, t, v, a, p, F
Kinematics in One Dimension
velocity wrt ground  vel person  vel train
 5 km/hr  80 km/hr
 85 km/hr
Velocities
Average velocity - over the trip, or distance, or time
Instantaneous velocity - right now speed
x
average speed: v 
t
instant speed:
x dx
v  lim

t  0 t
dt
Acceleration
How to express a change in velocity?
Again, two kinds of acceleration:
v
average acceleration: a 
t
v dv
instant acceleration: a  lim

t 0 t
dt
Kinematics defined by - x, t, v, a
x  displacement
t  time
v  velocity
x
v
t
x dx
v  lim

t  0 t
dt
a  acceleration
v
a
t
v dv
a  lim

t  0 t
dt
 d 2x 
 2 
 dt 
An automobile is moving along a straight highway, and
the driver puts on the brakes. If the initial velocity is
v1 = 15.0 m/s and it takes 5.0 s to slow to v2 = 5.0 m/s,
what is the car’s average acceleration?
From the definition for average acceleration:
v2  v1
a
t
5.0 m/s  15.0 m/s

5.0 s
2
 2.0 m/s
Motion at Constant Acceleration
kinematics - x, t, v, a
How are these related?
For simplicity, assume that the acceleration is constant:
a = const
Consider some
acceleration:
v
a
t
v  v0

t
v  v0  a t
The resulting
velocity:
v  v0  a t
x x  x0
v

t
t
For a constant
acceleration:
x  x0  v t
v  v0
v
2
 v0  v 
 v0  v0  at 
x  x0  
 t  x0  
t
2
 2 


Realize a
displacement:
x  x0  v0 t  12 a t 2
How about an equation of motion without time?
v  v0
x  x0  vt
v
2
 v  v0  v  v0 
 x0  


 2  a 
v 2  v02
 x0 
2a
v  v  2a  x  x0 
2
2
0
v  v0
t
a
v  v0  a t
a  const
 v  v0 
v 
t

0
0

 2 
2
1
x  x0  v0 t  2 a t
v  v  2a  x  x0 
2
2
0