01-Chapter 01 _ Units, Physical Quantities, And Vectors

Copyright Pearson Education Inc – Modified 8/15 by Scott Hildereth, Chabot College.

PowerPoint® Lectures forUniversity Physics, Thirteenth Edition

– Hugh D. Young and Roger A. Freedman

Physics PHU124 – FALL 2015Dr. Farid Amalou

CHAPTER 1

Units, Physical Quantities, and Vectors

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

Three KEYS for Chapter 1

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

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

– Dimensional Analysis

• Significant figures in calculations

– 6.696 x 104 miles/hour

– 67,000 miles hour

Three KEYS for Chapter 1

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

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

– Dimensional Analysis

• Significant figures in calculations

• Vectors (magnitude, direction, units) 5 m/s at 45°

What you MUST be able to do…

• Vectors & Vector mathematics

• vector componentsEx: v = velocity

• vx = v cosθ is the “x” component

• vy = v sinθ is the “y” component

• |v|2 = (vx)2 + (vy)2

5 m/s at 45°

3.54 m/s in “x”

3.54 m/s in “y”

What you MUST be able to do…

• Vectors & Vector mathematics

– vector componentsEx: v = velocity; vx = v cosθ

– unit vectors (indicating direction only)vx =

vy =

– Adding, subtracting, & multiplying vectors

Standards and units

• Length, mass, and time = three fundamentalquantities (“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 inkilograms, and time in seconds.

Unit consistency and conversions

• An equation must be dimensionally consistent. Terms to be added or equated must alwayshave 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

Unit consistency and conversions

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

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

5 meters/sec x 10 hour x (3600 sec/hour)= 180,000 meters = 180 km = ~ 2 x 102 km

Unit consistency and conversions

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

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

• NOT: 5 meters/sec x 10 kg = 50 Joules(velocity) x (mass) = (energy)

Unit prefixes

• Table 1.1 shows some larger and smaller units for the 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!

Measurement & Uncertainty

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).

Numbers ending in zero are ambiguous. Does the last zero mean uncertainty to a factor of 10, or just 1?

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 is commonly used in physics; it 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.

Measurement & Uncertainty

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 and significant figures

• Uncertainty should match measurement in the least precise digit:

–8.7 ± 0.1 centimeters

–8.70 ± 0.10 centimeters–8.709 ± 0.034 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?

Uncertainty and significant figures

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

Uncertainty and significant figures

• 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?

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? What uncertainty?

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

Conceptual Example: Significant figures

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

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

Conceptual Example: Significant figures

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

(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

Accuracy vs. Precision

Accuracy vs. Precision ?

Accuracy vs. Precision ?

Use least-squares fit to find line that minimizes deviation

Large error bars (uncertainty in

measurements) = not very precise…

Lots of data IMPROVES fit

and overall precision

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 vs. Precision

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! ☺

Accuracy vs. Precision

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!

Accuracy vs. Precision

Better Technique: Include uncertaintyYour 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!

Accuracy vs. Precision

Better Technique: Include uncertaintyYour 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!

Accuracy vs. Precision

•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.

Order of Magnitude: Rapid Estimating

Example: Volume of a lake

Volume = π x r2 x depth

= ~ 3 x 500 x 500 x 10

= ~75 x 105

= ~ 100 x 105

= ~ 107 cubic meters

Order of Magnitude: Rapid Estimating

Example: Volume of a lake

Volume = π x r2 x depth

= 7,853,981.634 cu. m

~ 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 textbook offers a systematic problem-solving strategywith techniques for setting up and solving problems efficiently and accurately.

Solving problems in physics

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

Solving problems in physics

• 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?

Solving problems in physics

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

Solving problems in physics

• 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

– BRIDGING Problem @ end of each chapter ***

Vectors and scalars

• A scalar quantity can be described by a single number, with some meaningful unit

• 4 oranges

• 20 miles

• 5 miles/hour

• 10 Joules of energy

• 9 Volts

Vectors and scalars

• A scalar quantity can be described by a single number with some meaningful unit

• A vector quantity has a magnitude and a direction in space, as well as some meaningful unit.

• 5 miles/hour North

• 18 Newtons in the “x direction”

• 50 Volts/meter down

Vectors and scalars

• A scalar quantity can be described by a single number with some meaningful unit

• A vector quantity has a magnitude and a direction in space, as well as some meaningful unit.

• To establish the direction, you MUST first have a coordinate system!

• Standard x-y Cartesian coordinates common

• Compass directions (N-E-S-W)

Drawing vectors• Draw a vector as a line with an arrowhead at its tip.

• The length of the line shows the vector’s magnitude.

• The direction of the line shows the vector’s directionrelative to a coordinate system (that should be indicated!)

x

y

z

5 m/sec at 30 degrees from the

x axis towards y in the xy plane

Drawing vectors• Vectors can be identical in magnitude, direction, and units,

but start from different places…

Drawing vectors• Negative vectors refer to direction relative to some standard

coordinate already established – not to magnitude.

Adding two vectors graphically

• Two vectors may be added graphically using either the head-to-tailmethod or the parallelogram method.

Adding two vectors graphically

• Two vectors may be added graphically using either the head-to-tailmethod or the parallelogram method.

Adding two vectors graphically

Adding more than two vectors graphically

• To add several vectors, use the head-to-tail method.

• The vectors can be added in any order.

Adding more than two vectors graphically—Figure 1.13

• To add several vectors, use the head-to-tail method.

• The vectors can be added in any order.

Subtracting vectors

• Reverse direction, and add normally head-to-tail…

Subtracting vectors

• Figure 1.14 shows how to subtract vectors.

Multiplying a vector by a scalar

• If c is a scalar, the product cA has magnitude |c|A.

→

Addition of two vectors at right angles• First add vectors graphically.

• Use trigonometry to find magnitude & direction of sum.

Addition of two vectors at right angles• Displacement (D) = √(1.002 + 2.002) = 2.24 km

• Direction φ = tan-1(2.00/1.00) = 63.4º East of North

Note how the final answer has THREE things!

• Answer: 2.24 km at 63.4 degrees East of North

• Magnitude (with correct sig. figs!)

Note how the final answer has THREE things!

• Answer: 2.24 km at 63.4 degrees East of North

• Magnitude (with correct sig. figs!)

• Units

Note how the final answer has THREE things!

• Answer: 2.24 km at 63.4 degrees East of North• Magnitude (with correct sig. figs!)

• Units

• Direction

Components of a vector

• Represent any vector by an x-component Ax and a y-component Ay.

• Use trigonometry to find the components of a vector: Ax = Acos θ and Ay = Asin θ, where θ is measured from the +x-axis toward the +y-axis.

Positive and negative components

• The components of a vector can be positive or negative numbers.

Finding components

• We can calculate the components of a vector from its magnitude and direction.

Calculations using components• We can use the components of a vector to find its magnitude

and direction:

• We can use the components of a set of vectors to find the components of their sum:

2 2     and      tanθ= + = yx y

x

AA A A A

,   = + + + = + + +L Lx x x x y y y yR A B C R A B C

Adding vectors using their components

Unit vectors

• A unit vector has a magnitude of 1 with no units.

• The unit vector î points in the +x-direction, points in the +y-direction, and points in the +z-direction.

• Any vector can be expressed in terms of its components as A =Axî+ Ay + Az .$j$ $k$

$j$$k$

→

The scalar product

The scalar product of two vectors (the “dot product”) is

A · B = ABcosφ

The scalar product

The scalar product of two vectors (the “dot product”) is

A · B = ABcosφ

Useful for

•Work (energy) required or released as force is applied over a distance (4A)

•Flux of Electric and Magnetic fields moving through surfaces and volumes in space (4B)

Calculating a scalar product

By components, A · B = AxBx + AyBy + AzBz

Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130°

Calculating a scalar product

By components, A · B = AxBx + AyBy + AzBz

Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130°

Ax = 4.00 cos 53 = 2.407

Ay = 4.00 sin 53 = 3.195

Bx = 5.00 cos 130 = -3.214

By = 5.00 sin 130 = 3.830

AxBx + AyBy = 4.50 meters

A · B = ABcosφ = (4.00)(5.00) cos(130-53) = 4.50 meters2

The vector product

•The vector product (“cross product”) A x B of two vectors is a vector

•Magnitude = AB sin φ

•Direction = orthogonal (perpendicular) to A and B, using the “Right Hand Rule”

A

B

A x B

x

y

z

The vector cross product

The cross product of two vectors is

A x B (with magnitude ABsinφ)

Useful for

•Torque from a force applied at a distance away from an axle or axis of rotation (4A)

•Calculating dipole moments and forces from Magnetic Fields on moving charges (4B)

The vector product

• The vector product (“cross product”) of two vectors has magnitude

and the right-hand rule gives its direction.

| |  sinφ× =r r

ABA B