Units, Physical Quantities, and Vectors

Units, Physical Quantities, and Vectors
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 3

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° 4

What you MUST be able to do…
Vectors & Vector mathematics vector components Ex: v = velocity vx = v cosq is the “x” component vy = v sinq is the “y” component |v|2 = (vx)2 + (vy)2 5 m/s at 45° 3.54 m/s in “y” 3.54 m/s in “x” 5

What you MUST be able to do…
Vectors & Vector mathematics vector components Ex: v = velocity; vx = v cosq unit vectors (indicating direction only) vx = vy = Adding, subtracting, & multiplying vectors 6

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.

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

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 5 meters/sec x 10 hour x (3600 sec/hour) = 180,000 meters = 180 km = ~ 2 x 102 km 9

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

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. Figure 1-2. Caption: Measuring the width of a board with a centimeter ruler. The uncertainty is about ±1 mm.

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!

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 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

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: = 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 Figure 1-3. Caption: These two calculators show the wrong number of significant figures. In (a), 2.0 was divided by 3.0. The correct final result would be In (b), 2.5 was multiplied by 3.2.The correct result is 8.0.

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 x has three; and x 104 has four.

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. Figure 1-2. Caption: Measuring the width of a board with a centimeter ruler. The uncertainty is about ±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 20

Uncertainty should match measurement in the least precise digit: 8.7  0.1 centimeters 8.70  0.10 centimeters 8.709  centimeters 8  1 centimeters Not… 8.7 +/ cm 21

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. 23

As this train mishap illustrates, even a small percent error can have spectacular results! 24

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? Figure 1-4. Caption: Example 1–1. A protractor used to measure an angle. Response: (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two 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) Figure 1-4. Caption: Example 1–1. A protractor used to measure an angle. Response: (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two 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? Figure 1-4. Caption: Example 1–1. A protractor used to measure an angle. Response: (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two 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? , but to two sig figs, 0.87! Figure 1-4. Caption: Example 1–1. A protractor used to measure an angle. Response: (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two significant figures.

Key Concepts for the Day!
Class Calendar Mastering Physics Intro Assignment Results Precision vs. Accuracy Vectors

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 ? Use least-squares fit to find line that minimizes deviation Lots of data IMPROVES fit and overall precision Large error bars (uncertainty in measurements) = not very precise…

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 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 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 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
Accuracy vs. Precision Better Technique: Include uncertainty Your experiment produces m/sec2 +/ 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
Accuracy vs. Precision Better Technique: Include uncertainty Your experiment produces 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!

established by uncertainty in a measurement
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. Figure 1-7. Caption: Example 1–5. (a) How much water is in this lake? (Photo is of one of the Rae Lakes in the Sierra Nevada of California.) (b) Model of the lake as a cylinder. [We could go one step further and estimate the mass or weight of this lake. We will see later that water has a density 1000 kg/m3, of so this lake has a mass of about (103 kg/m3)(107 m3) ≈ 1010 kg, which is about 10 billion kg or 10 million metric tons. (A metric ton is 1000 kg, about 2200 lbs, slightly larger than a British ton, 2000 lbs.)] Answer: The volume of the lake is about 107 m3.

Example: Volume of a lake Volume = p x r2 x depth = ~ 3 x 500 x 500 x 10 = ~75 x 105 = ~ 100 x 105 = ~ 107 cubic meters Figure 1-7. Caption: Example 1–5. (a) How much water is in this lake? (Photo is of one of the Rae Lakes in the Sierra Nevada of California.) (b) Model of the lake as a cylinder. [We could go one step further and estimate the mass or weight of this lake. We will see later that water has a density 1000 kg/m3, of so this lake has a mass of about (103 kg/m3)(107 m3) ≈ 1010 kg, which is about 10 billion kg or 10 million metric tons. (A metric ton is 1000 kg, about 2200 lbs, slightly larger than a British ton, 2000 lbs.)] Answer: The volume of the lake is about 107 m3.

Example: Volume of a lake Volume = p x r2 x depth = 7,853, cu. m ~ 107 cubic meters Figure 1-7. Caption: Example 1–5. (a) How much water is in this lake? (Photo is of one of the Rae Lakes in the Sierra Nevada of California.) (b) Model of the lake as a cylinder. [We could go one step further and estimate the mass or weight of this lake. We will see later that water has a density 1000 kg/m3, of so this lake has a mass of about (103 kg/m3)(107 m3) ≈ 1010 kg, which is about 10 billion kg or 10 million metric tons. (A metric ton is 1000 kg, about 2200 lbs, slightly larger than a British ton, 2000 lbs.)] Answer: The volume of the lake is about 107 m3.

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!) Figure 1-8. Caption: Example 1–6. Micrometer used for measuring small thicknesses. Answer: Measure the thickness of 100 pages. You should find that the thickness of a single page is about 6 x 10-2 mm.

Solving problems in physics
The textbook offers a systematic problem-solving strategy with techniques for setting up and solving problems efficiently and accurately. 47

Step 1: Identify relevant concepts, variables, what is known, what is needed, what is missing. 48

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? 49

Step 3: Execute the Solution, and EVALUATE your answer! Are the units right? Is it the right order of magnitude? Does it make SENSE? 50

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 end of each chapter *** 51

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

To establish the direction, you MUST first have a coordinate system!
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 direction relative 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… 56

Drawing vectors Negative vectors refer to direction relative to some standard coordinate already established – not to magnitude. 57

Adding two vectors graphically
Two vectors may be added graphically using either the head-to-tail method or the parallelogram method.

Two vectors may be added graphically using either the head-to-tail method or the parallelogram method. 59

60

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. 62

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) = √( ) = 2.24 km Direction f = tan-1(2.00/1.00) = 63.4º East of North 67

Note how the final answer has THREE things!
Answer: 2.24 km at 63.4 degrees East of North Magnitude (with correct sig. figs!) 68

Answer: 2.24 km at 63.4 degrees East of North Magnitude (with correct sig. figs!) Units 69

Answer: 2.24 km at 63.4 degrees East of North Magnitude (with correct sig. figs!) Units Direction 70

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:

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 . 

The scalar product The scalar product of two vectors (the “dot product”) is A · B = ABcosf

The scalar product The scalar product of two vectors (the “dot product”) is A · B = ABcosf 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) 78

Calculating a scalar product
By components, A · B = AxBx + AyBy + AzBz Example: A = °, B = °

Calculating a scalar product
By components, A · B = AxBx + AyBy + AzBz Example: A = °, B = ° Ax = 4.00 cos 53 = 2.407 Ay = 4.00 sin 53 = 3.195 Bx = 5.00 cos 130 = By = 5.00 sin 130 = 3.830 AxBx + AyBy = 4.50 meters A · B = ABcosf = (4.00)(5.00) cos(130-53) = 4.50 meters2 80

The vector product The vector product (“cross product”) A x B of two vectors is a vector Magnitude = AB sin f Direction = orthogonal (perpendicular) to A and B, using the “Right Hand Rule” x y z B A A x B

The vector cross product
The cross product of two vectors is A x B (with magnitude ABsinf) 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) 82

The vector product The vector product (“cross product”) of two vectors has magnitude and the right-hand rule gives its direction. 83

Units, Physical Quantities, and Vectors

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