Units, Physical Quantities, and Vectors - Bancroft School · Copyright © 2012 Pearson Education Inc. Calculating the vector product—Figure 1.32 ... Author: Simone Kang Created

Copyright © 2012 Pearson Education Inc.

PowerPoint® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman

Lectures by Wayne Anderson

Chapter 1

Units, Physical Quantities, and Vectors


Goals for Chapter 1 •  To learn three fundamental quantities of physics and the

units to measure them

•  To keep track of significant figures in calculations

•  To understand vectors and scalars and how to add vectors graphically

•  To determine vector components and how to use them in calculations

•  To understand unit vectors and how to use them with components to describe vectors

•  To learn two ways of multiplying vectors


The nature of physics

• Physics is an experimental science in which physicists seek patterns that relate the phenomena of nature.

• The patterns are called physical theories.

• A very well established or widely used theory is called a physical law or principle.


Solving problems in physics •  A problem-solving strategy offers techniques for setting up

and solving problems efficiently and accurately.


Why use units

•  This table is 1000 long.

•  This table is 1 long.


Why have different units

•  The speed of a car is 26 m/s.

•  The speed of a car is 26,000 mm/s.

•  A pencil is 15 cm long.

•  A pencil is 0.00015 km long.


Standards and units

•  Length, time, and mass are three fundamental quantities of physics.

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

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


What is one meter

•  The meter is the basic unit of length in the SI system of units. The meter is defined to be the distance light travels through a vacuum in exactly 1/299792458 seconds.

•  U.S. PROTOTYPE METER BAR


What is one kilogram

•  It is defined as the mass of a particular international prototype made of platinum-iridium and kept at the International Bureau of Weights and Measures. It was originally defined as the mass of one liter (10-3 cubic meter) of pure water.

•  France holds the world's standard for the kilogram weight. It's a cylinder-shaped object that has dictated all other kilogram weights for 130 years now.


What is one second

• The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.


Unit prefixes

• Table 1.1 shows some larger and smaller units for the fundamental quantities.


More prefix for powers of ten yotta- (Y-) 1024 1 septillion zetta- (Z-) 1021 1 sextillion exa- (E-) 1018 1 quintillion peta- (P-) 1015 1 quadrillion tera- (T-) 1012 1 trillion giga- (G-) 109 1 billion mega- (M-) 106 1 million kilo- (k-) 103 1 thousand hecto- (h-) 102 1 hundred deka- (da-) 10 1 ten deci- (d-) 10-1 1 tenth centi- (c-) 10-2 1 hundredth milli- (m-) 10-3 1 thousandth micro- (µ-) 10-6 1 millionth nano- (n-) 10-9 1 billionth pico- (p-) 10-12 1 trillionth femto- (f-) 10-15 1 quadrillionth atto- (a-) 10-18 1 quintillionth zepto- (z-) 10-21 1 sextillionth yocto- (y-) 10-24 1 septillionth


British System vs. SI Length kilometer (km) = 1,000 m 1 km = 0.621 mi 1 mi = 1.609 km meter (m) = 100 cm 1 m = 3.281 ft 1 ft = 0.305 m centimeter (cm) = 0.01 m 1 cm = 0.394 in. 1 in. = 2.540 cm millimeter (mm) = 0.001 m 1 mm = 0.039 in. micrometer (µm) = 0.000 001 m nanometer (nm) = 0.000 000 001 m Area square kilometer (km2) = 100 hectares 1 km2 = 0.386 mi2 1 mi2 = 2.590 km2 hectare (ha) = 10,000 m2 1 ha = 2.471 acres 1 acre = 0.405 ha square meter (m2) = 10,000 cm2 1 m2 = 10.765 ft2 1 ft2 = 0.093 m2 square centimeter (cm2) = 100 mm2 1 cm2 = 0.155 in.2 1 in.2 = 6.452 cm2 Volume liter (L) = 1,000 mL = 1 dm3 1 L = 1.057 fl qt 1 fl qt = 0.946 L milliliter (mL) = 0.001 L = 1 cm3 1 mL = 0.034 fl oz 1 fl oz = 29.575 mL microliter (µL) = 0.000 001 L Mass kilogram (kg) = 1,000 g 1 kg = 2.205 lb 1 lb = 0.454 kg gram (g) = 1,000 mg 1 g = 0.035 oz 1 oz = 28.349 g milligram (mg) = 0.001 g


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.”)

•  Always carry units through calculations. •  Convert to standard units as necessary. (Follow Problem-Solving

Strategy 1.2) •  Follow Examples 1.1 and 1.2.


Uncertainty and significant figures—Figure 1.7 •  The uncertainty of a measured quantity

is indicated by its number of significant figures.

•  For multiplication and division, the answer can have no more significant figures than the smallest number of significant figures in the factors.

•  For addition and subtraction, the number of significant figures is determined by the term having the fewest digits to the right of the decimal point.

•  Refer to Table 1.2, Figure 1.8, and Example 1.3.

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


Ways to express accuracy

•  56.47 ± 0.02

•  47 ± 10% (fractional/percent error, fractional/percent uncertainty)


Estimates and orders of magnitude

•  An order-of-magnitude estimate of a quantity gives a rough idea of its magnitude.

•  Can you count 1 million in your life?

•  Follow Example 1.4.


Vectors and scalars

• A scalar quantity can be described by a single number.

• A vector quantity has both a magnitude and a direction in space.

•  In this book, a vector quantity is represented in boldface italic type with an arrow over it: A.

• The magnitude of A is written as A or |A|.

è

è è


Drawing vectors—Figure 1.10 •  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.

•  Figure 1.10 shows equal-magnitude vectors having the same direction and opposite directions.


Adding two vectors graphically—Figures 1.11–1.12 •  Two vectors may be added graphically using either the parallelogram

method or the head-to-tail method.


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

•  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.

•  Figure 1.15 illustrates multiplication of a vector by a positive scalar and a negative scalar.

è


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

•  Then use trigonometry to find the magnitude and direction of the sum.



Components of a vector—Figure 1.17 •  Adding vectors graphically provides limited accuracy. Vector

components provide a general method for adding vectors.

•  Any vector can be represented 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—Figure 1.18

• The components of a vector can be positive or negative numbers, as shown in the figure.


Finding components—Figure 1.19 •  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:

•  Refer to Problem-Solving Strategy 1.3.

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—Figure 1.22

•  Follow Examples 1.7 and 1.8.


Unit vectors—Figures 1.23–1.24

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

•  The unit vector î points in the +x-direction, ĵ 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 .

•  Follow Example 1.9. j$j$

k$k$

k$k$

è


The scalar product—Figures 1.25–1.26

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

•  Figures 1.25 and 1.26 illustrate the scalar product.


Calculating a scalar product

[Insert figure 1.27 here]

•  In terms of components, •  Example 1.10 shows how to calculate a scalar product in two

ways.


Finding an angle using the scalar product

•  Example 1.11 shows how to use components to find the angle between two vectors.


The vector product—Figures 1.29–1.30 •  The vector

product (“cross product”) of two vectors has magnitude

and the right-hand rule gives its direction. See Figures 1.29 and 1.30.

|rA×rB | = ABsinφ


Calculating the vector product—Figure 1.32

•  Use ABsinΦ to find the magnitude and the right-hand rule to find the direction.

•  Refer to Example 1.12.

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Units, Physical Quantities, and Vectors - Bancroft School · Copyright © 2012 Pearson Education Inc. Calculating the vector product—Figure 1.32 ... Author: Simone Kang Created