scott hildreth – chabot college – adapted from wiley fundamentals of physics 10e; pearson...

Scott Hildreth – Chabot College – Adapted from Wiley Fundamentals of Physics 10e; Pearson University Physics 13e

Chapter 1Measurement

Fundamental Quantities in Physics

Units & Conversion

Three KEYS for Chapter 1

• Fundamental quantities in physics (length, mass, time)

– Units (meters, kilograms, seconds...)

– Dimensional Analysis

• Force = kg meter/sec2

• Power = Force x Velocity

= kg m2/sec3





• Significant figures in calculations

– 6.696 x 104 miles/hour

– 67,000 miles hour





• Significant figures in calculations

• Estimation (order of magnitude ~10#)

Standards and units

• Length, mass, and time = three fundamental quantities (“dimensions”) of physics.

• The SI (Système International) is the most widely used system of units.

– Meeting ISO standards are mandatory for some industries. Why?

• In SI units, length is measured in meters, mass in kilograms, and time in seconds.

http://www.iso.org/iso/home/standards.htm

http://www.iso.org/iso/home/standards.htm

http://www.iso.org/iso/10goodthings.pdf

http://www.iso.org/iso/10goodthings.pdf

Converting Units

•A conversion factor is

•A ratio of units equal to 1

•Used to convert between units

• Units obey same algebraic rules as variables & numbers

Converting Units

km

Converting Units

km

1000 m = 1 km

Multiplying by 1 doesn’t change the overall value, just the units.

Converting Units

km

Converting Units

km

cm

Unit consistency and conversions

• An equation must be dimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”)

• OK: 5 meters/sec x 10 hours =~ 2 x 102 km

(distance/time) x (time) = distance




5 meters/sec x 10 hour x (3600 sec/hour)

= 180,000 meters = 180 km = ~ 2 x 102 km




• NOT: 5 meters/sec x 10 kg = 50 Joules

(velocity) x (mass) = (energy)

Unit prefixes

• Larger & smaller units for fundamental quantities.

• Learn these – and prefixes like Mega, Tera, Pico, etc.!

• Skip Ahead to Slide 24 – Sig Fig Example

Measurement & Uncertainty

No measurement is exact; there is always some uncertainty due to limited instrument accuracy and difficulty reading results.

• The precision – and also uncertainty - of a measured quantity is indicated by its number of significant figures.

–Ex: 8.7 centimeters

• 2 sig figs

• Specific rules for significant figures exist

• In online homework, sig figs matter!

• In exams, sig figs matter!!


Significant Figures

Number of significant figures = number of “reliably known digits” in a number.

Often possible to tell # of significant figures by the way the number is written:

• 23.21 cm = four significant figures.

• 0.062 cm = two significant figures (initial zeroes don’t count).

Significant Figures

• Significant figures are not decimal places

0.00356 has 5 decimal places, but just 3 significant figures

Generally, round to the least number of significant figures of the given data 25 x 18 → 2 significant figures; 25 x 18975 → still 2 Round up for 5+ (13.5 → 14, but 13.4 → 13)

In general, trailing zeros are NOT significant

In other words, 3000 may have 4 significant figures

but usually 3000 will have only 1 significant figure!

Numbers ending in zero are ambiguous.

Does the last zero mean uncertainty to a factor of 10, or just 1?

Significant Figures

Numbers ending in zero are ambiguous

Is 20 cm precise to 10 cm, or 1? We need rules!

• 20 cm = one significant figure(trailing zeroes don’t count w/o decimal point)

• 20. cm = two significant figures(trailing zeroes DO count w/ decimal point)

• 20.0 cm = three significant figures

Significant Figures

Rules for Significant Figures

•When multiplying or dividing numbers, or using functions, result has as many sig figs as term with fewest (the least precise).

•ex: 11.3 cm x 6.8 cm = 77 cm.

•When adding or subtracting, answer is no more precise than least precise number used.

• ex: 1.213 + 2 = 3, not 3.213!

Significant Figures

•Calculators will not give right # of sig figs; usually give too many but sometimes give too few (especially if there are trailing zeroes after a decimal point).

•top image: result of 2.0/3.0

•bottom image: result of 2.5 x 3.2

Scientific Notation

•Scientific notation commonly used

•Uses powers of 10 to write large & small numbers

Scientific Notation

•Scientific notation allows the number of significant figures to be clearly shown.

•Ex: cannot easily tell how many significant figures in “36,900”.

•Clearly 3.69 x 104 has three and 3.690 x 104 has four!

Remember trailing zeroes DO count with a decimal point (always in Scientific Notation!)


No measurement is exact; there is always some uncertainty due to limited instrument accuracy and difficulty reading results.

Photo illustrates this – it would be difficult to measure the width of this board more accurately than ± 1 mm.

Uncertainty and significant figures

• Every measurement has uncertainty

–Ex: 8.7 cm (2 sig figs)

• “8” is (fairly) certain

• 8.6? 8.8?

• 8.71? 8.69?

• Good practice – include uncertainty with every measurement!

–8.7 0.1 meters


• Uncertainty should match measurement in the least precise digit:

–8.7 0.1 centimeters



–8 1 centimeters

• Not…

–8.7 +/- 0.034 cm

Relative Uncertainty

•Relative uncertainty: a percentage, the ratio of uncertainty to measured value, multiplied by 100.

•ex. Measure a phone to be 8.8 ± 0.1 cm

What is the relative uncertainty in this measurement?


• Physics involves approximations; these can affect the precision of a measurement.


• As this train mishap illustrates, even a small percent error can have spectacular results!

Conceptual Example: Significant figures

Using a protractor, you measure an angle to be 30°.

(a) How many significant figures should you quote in this measurement?



(a) How many significant figures should you quote in this measurement? What uncertainty?

2 sig figs! (30. +/- 1 degrees or 3.0 x 101 +/- 1 degrees)



(b) What result would a calculator give for the cosine of this result? What should you report?



(b) What result would a calculator give for the cosine of this result? What should you report?

0.866025403, but to two sig figs, 0.87!

1-3 Accuracy vs. Precision

Accuracy is how close a measurement comes to the true value.

ex. Acceleration of Earth’s gravity = 9.81 m/sec2

Your experiment produces 10 ± 1 m/sec2

• You were accurate! How accurate? Measured by ERROR.

• |Actual – Measured|/Actual x 100%

• | 9.81 – 10 | / 9.81 x 100% = 1.9% Error

Accuracy vs. Precision

•Accuracy is how close a measurement comes to the true value

• established by % error

•Precision is a measure of repeatability of the measurement using the same instrument.

• established by uncertainty in a measurement

• reflected by the # of significant figures

Accuracy vs. Precision Example

•Example:

You measure the acceleration of Earth’s gravitational force in the lab, which is accepted to be 9.81 m/sec2

• Your experiment produces 8.334 m/sec2

•Were you accurate? Were you precise?


Accuracy is how close a measurement comes to the true value. (established by % error)

ex. Your experiment produces 8.334 m/sec2

for the acceleration of gravity (9.81 m/sec2)

Accuracy: (9.81 – 8.334)/9.81 x 100% = 15% error

Is this good enough? Only you (or your boss/customer) know for sure!


Precision is the repeatability of the measurement using the same instrument.

ex. Your experiment produces 8.334 m/sec2

for the acceleration of gravity (9.81 m/sec2)

Precision indicated by 4 sig figs

Seems (subjectively) very precise – and precisely wrong!


Better Technique: Include uncertainty

Your experiment produces

8.334 m/sec2 +/- 0.077 m/sec2

Your relative uncertainty is

.077/8.334 x 100% = ~1%

But your error was ~ 15%

NOT a good result!


Better Technique: Include uncertainty

Your experiment produces

8.3 m/sec2 +/- 1.2 m/sec2

Your relative uncertainty is

1.2 / 8.3 x 100% = ~15%

Your error was still ~ 15%

Much more reasonable a result!


•Precision is a measure of repeatability of the measurement using the same instrument.

• established by uncertainty in a measurement

• reflected by the # of significant figures

• improved by repeated measurements!

•Statistically, if each measurement is independent

• make n measurements (and n> 10)

•Improve precision by √(n-1)

• Make 10 measurements, % uncertainty ~ 1/3

1-6 Order of Magnitude: Rapid Estimating

Quick way to estimate calculated quantity:

• round off all numbers in a calculation to one significant figure and then calculate.

• result should be right order of magnitude

• expressed by rounding off to nearest power of 10

• 104 meters

• 108 light years

Order of Magnitude: Rapid Estimating

Example: Volume of a lake

Estimate how much water there is in a particular lake, which is roughly circular, about 1 km across, and you guess it has an average depth of about 10 m.



Volume = Area x depth= (x r2) x depth

= ~ 3 x 500 x 500 x 10

= ~75 x 105

= ~ 100 x 105

= ~ 107 cubic meters



Volume = (x r2) x depth

= 7,853,981.634 cu. m

But…. Round to power of 10 for Order of Mag:

So ~ 107 cubic meters

1-6 Order of Magnitude: Rapid Estimating

Example: Thickness of a page.

Estimate the thickness of a page of your textbook.

(Hint: you don’t need one of these!)

Solving problems in physics

• The online system offers a HUGE array of additional resources to help you visualize how to solve problems


• The textbook sample problems are IMPORTANT

Solving problems in physics – Step by Step!

• Step 1: Identify KEY IDEAS, relevant concepts, variables, what is known, what is needed, what is missing.


• Step 2: Set up the Problem – MAKE a SKETCH, label it, act it out, model it, decide what equations might apply. What units should the answer have? What value?


• Step 3: Execute the Solution, and EVALUATE your answer! Are the units right? Is it the right order of magnitude? Does it make SENSE?


• Good problems to gauge your learning

– “Test your Understanding” Questions throughout the book

– Conceptual “Clicker” questions linked online

– “Two dot” problems in the chapter

• Good problems to review before exams

– Checkpoints along the way

– ODD problems with answers in the back

– Exam reviews published online

scott hildreth – chabot college – adapted from wiley fundamentals of physics 10e; pearson...

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