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 & limitationsWays to circumvent

Measurement standards

Many units to match the dimension of an itemor 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

22

2

2 2

13.7 cm 0.1cm (0.7%)

Area2

4

147.4113813 cm 2.1cm

d

dr

d

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 PhysicsWorkshop 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 more importantimportant than either problem solving or the comprehensive transmission of descriptive knowledge about the enterprise of physics.

• There were two major reasonstwo major reasons for the emphasis on inquiry skills based on real experience.

• FirstFirst, the majoritymajority of students enrolled in introductory physics at both the high school and college level do not have sufficient do not have sufficient concrete experienceconcrete 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 experiencerelate concrete experience to scientific scientific explanation. explanation.

• A second 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 acquire methods for independent investigationinvestigation to be implemented as needed.

• Although lectureslectures and demonstrations demonstrations are useful alternatives to reading reading for transmitting information and teaching teaching specific skills, they are unproved as vehicles for helping students learn how to thinkthink, conduct scientific inquiryconduct scientific inquiry, or acquire real experience with natural acquire real experience with natural phenomena.phenomena.

• The time now spent by students passively listening to lectures is better spent in direct direct inquiryinquiry and discussiondiscussion with peers.

• Many educators believe that peers are often more helpfulmore helpful than instructors in facilitating original thinking and problem solving on the part of students.

Statistical Measures

• Systematic errorsSystematic 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 errorsRandom 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 randomrandom 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

B

1 Tasteeos (g)

2 40.8

3 42.9

4 42.3

5 41.9

6 42.5

7 40.5

8 41.816666667

9 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 L2 L3

40.8

42.9

42.3

41.9

42.5

40.5

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

•x = 41.81666667

x=250.9

x2=10496.45

•Sx=.9641922353

•σx=.8801830618•n=6

Where is the average?

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 “reliablereliable” central measure of the data is the mean (averagemean (average).

• The first moment of the distribution, the standard deviationstandard deviation, is related to the probability that each measurement is close to the mean.

• Standard deviationStandard deviation tells us how close anclose an additional measurement would come 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 (gaussiangaussian) distribution on a histogram. Even if it doesn’t, it is convenient to model the data as gaussiangaussian to calculate the Std Dev.Std Dev.

• Another reliable measure for the data is the standard deviation of the mean (SDM).standard deviation of the mean (SDM). This expresses the probability that the mean can vary. The SDMSDM is gaussian for large sample sizes.

• There are higher moments of the distribution which are informative in some situations (e.g., skew, kurtosis).

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.

Behold the Histogram!

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:

= x x x x x

NN1 2 3 4

(C.1)

ri = – xi

SD = sd = ( )

( )

r r r r r

NN1

22

23

24

2 2

1

= r

Ni2

1

( )

Here the symbol means “sum the terms i = 1 to i = N.”

Consider the definition of the mean and the std dev (standard deviation).

A test of significance is if any new data is beyond the 95% (“2σ” or two standard deviations) level.

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

Estimating Volume

2

2

3 6 3

E

E

Vol =

3 (5.0 2 m) (10 m)

8 6 m (8 10 m )

r h

Estimating Speed

According to a rule-of-thumb, every five seconds betweena lightning flash and the following thunder gives the distanceof the storm in miles. Assuming that the flash of light arrivesin essentially no time at all, estimate the speed of sound in m/sfrom this rule.

1 mi 1609 m 322 m/s

5 s 1 mi

Kinematics in One Dimension

MECHANICS comes in two parts:

kinematics: motion (displacement, time, velocity)x, t, v, a

dynamics: motion and forcesx, t, v, a, p, F

Kinematics in One Dimension

person trainvelocity wrt ground vel vel

5 km/hr 80 km/hr

85 km/hr

Velocities

0

average speed:

instant speed: limt

xv

t

x dxv

t dt

Average velocity - over the trip, or distance, or time

Instantaneous velocity - right now speed

Acceleration

How to express a change in velocity?

Again, two kinds of acceleration:

0

average acceleration:

instant acceleration: limt

va

t

v dva

t dt

Kinematics defined by - x, t, v, a

x displacementt timev velocity

0lim

t

xv

tx dx

vt dt

a acceleration

2

20lim

t

va

t

v dv d xa

t dt dt

An automobile is moving along a straight highway, andthe driver puts on the brakes. If the initial velocity isv1 = 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:

2 1

2

5.0 m/s 15.0 m/s

5.0 s

2.0 m/s

v va

t

Motion at Constant Acceleration

kinematics - x, t, v, a

How are these related?

For simplicity, assume that the acceleration is constant:

a = const

0

0

0

va

tv v

t

v v a t

v v a t

Consider someacceleration:

The resultingvelocity:

0

00

0 0 00 0

210 0 2

2

2 2

x xxv

t t

v vx x v t v

v v v v atx x t x t

x x v t a t

For a constantacceleration:

Realize adisplacement:

0 00

0 00

2 20

0

2 20 0

2

2

2

2

v v v vx x vt v t

av v v v

xa

v vx

a

v v a x x

How about an equation of motion without time?

0

00

210 0 2

2 20 0

02

2

v v a t a const

v vv t

x x v t a t

v v a x x