College Physics

Chapter 1Introduction

Fundamental Quantities and Their Dimension Length [L] Mass [M] Time [T]

other physical quantities can be constructed from these three

Units To communicate the result of a

measurement for a quantity, a unit must be defined

Defining units allows everyone to relate to the same fundamental amount

Systems of Measurement

Standardized systems agreed upon by some authority,

usually a governmental body

SI -- Systéme International agreed to in 1960 by an international

committee

Length

Units SI – meter, m [L]

Defined in terms of a meter – the distance traveled by light in a vacuum during a given time

Mass Units

SI – kilogram, kg [M]

Defined in terms of kilogram, based on a specific cylinder kept at the International Bureau of Weights and Measures

Time

Units seconds, s [T]

Defined in terms of the oscillation of radiation from a cesium atom

Approximate Values Various tables in the text show

approximate values for length, mass, and time Note the wide range of values Lengths – Table 1.1 Masses – Table 1.2 Time intervals – Table 1.3

Prefixes Prefixes correspond to powers of

10 Each prefix has a specific name Each prefix has a specific

abbreviation See table 1.4

Structure of Matter

Matter is made up of molecules the smallest division that is

identifiable as a substance

Molecules are made up of atoms correspond to elements

More structure of matter Atoms are made up of

nucleus, very dense, contains protons, positively charged, “heavy” neutrons, no charge, about same mass as

protons protons and neutrons are made up of quarks

orbited by electrons, negatively charges, “light”

fundamental particle, no structure

Structure of Matter

Dimensional Analysis Technique to check the

correctness of an equation Dimensions (length, mass, time,

combinations) can be treated as algebraic quantities add, subtract, multiply, divide

Both sides of equation must have the same dimensions

Dimensional Analysis, cont. Cannot give numerical factors: this

is its limitation Dimensions of some common

quantities are listed in Table 1.5

ExampleA shape that covers an area A and has a uniform height h has a volume V = Ah. Show that V = Ah is dimensionally correct.

3[ ] LV 2[ ] LA [ ] Lh

3 2L L L [ ] [ ][ ]V A h

V Ah

The units of volume, area and height are:

We then observe that

or

Thus, the equation is dimensionally correct.

Example

Suppose that the displacement of an object is related to time according to the expression x = Bt2. What are the dimensions of B?

2x Bt

2

xB

t

2

[ ][ ]

[ ]

xB

t

2

L

T

From

we find that

Thus, B has units of

Uncertainty in Measurements There is uncertainty in every

measurement, this uncertainty carries over through the calculations need a technique to account for this

uncertainty We will use rules for significant figures

to approximate the uncertainty in results of calculations

Significant Figures A significant figure is one that is reliably

known All non-zero digits are significant Zeros are significant when

between other non-zero digits after the decimal point and another

significant figure can be clarified by using scientific notation

Operations with Significant Figures Accuracy – number of significant figures When multiplying or dividing two or

more quantities, the number of significant figures in the final result is the same as the number of significant figures in the least accurate of the factors being combined

Operations with Significant Figures, cont. When adding or subtracting, round the

result to the smallest number of decimal places of any term in the sum

If the last digit to be dropped is less than 5, drop the digit

If the last digit dropped is greater than or equal to 5, raise the last retained digit by 1

The speed of light is now defined to be 2.99792458 × 108 m/s. Express the speed of light to (a) three significant figures, (b) five significant figures, and (c) seven significant figures.

Example

82.997 924 574 10 m sc 8

3.00 10 m s

8 2.997 9 10 m s

8 2.997 924 10 m s

(a) Rounded to 3 significant figures: c =

(b) Rounded to 5 significant figures: c =

(c) Rounded to 7 significant figures: c =

A farmer measures the perimeter of a rectangular field. The length of each long side of the rectangle is found to be 38.44 m, and the length of each short side is found to be 19.5 m. What is the perimeter of the field?

Example

38.44 m 19.5 m 38.44 m 19.5 m 115.88 m

115.9 m

19.5 m

The distance around is

but this answer must be rounded to

because the distance

carries information to only one place past the decimal.

Conversions When units are not consistent, you may

need to convert to appropriate ones Units can be treated like algebraic

quantities that can “cancel” each other See the inside of the front cover for an

extensive list of conversion factors Example:

2.5415.0 38.1

1

cmin cm

in

A rectangular building lot measures 100 ft by 150 ft. Determine the area of this lot in square meters (m2).

Example

2

4 2 1 m100 ft 150 ft 1.50 10 ft

3.281 ftA w

3 21.39 10 m

One cubic centimeter (1.0 cm3) of water has a mass of 1.0 × 10–3 kg. Determine the mass of 1.0 m3 of water.

Example

33

3

32 3 3 3

3

1.0 10 kgmass density volume 1.0 m

1.0 cm

kg 10 cm1.0 10 1.0 m 1.0 10 kg

cm 1 m

Examples of various units measuring a quantity

Order of Magnitude Approximation based on a number

of assumptions may need to modify assumptions if

more precise results are needed Order of magnitude is the power of

10 that applies

An automobile tire is rated to last for 50 000 miles. Estimate the number of revolutions the tire will make in its lifetime.

Example

750 000 mi 5280 ft mi 1 rev 8 ft 3 10 rev , or

7~10 rev

A reasonable guess for the diameter of a tire might be 2.5 ft, with a circumference of about 8 ft. Thus, the tire would make

Coordinate Systems Used to describe the position of a

point in space Coordinate system consists of

a fixed reference point called the origin

specific axes with scales and labels instructions on how to label a point

relative to the origin and the axes

Types of Coordinate Systems Cartesian Plane polar

Cartesian coordinate system

Also called rectangular coordinate system

x- and y- axes Points are labeled

(x,y)

Plane polar coordinate system Origin and

reference line are noted

Point is distance r from the origin in the direction of angle , from reference line

Points are labeled (r,)

Trigonometry Review

sin

cos

tan

opposite side

hypotenuse

adjacent side

hypotenuse

opposite side

adjacent side

More Trigonometry Pythagorean Theorem

To find an angle, you need the inverse trig function for example,

Be sure your calculator is set appropriately for degrees or radians

2 2 2r x y

1sin 0.707 45

A point is located in a polar coordinate system by the coordinates r = 2.5 m and θ = 35°. Find the x- and y-coordinates of this point, assuming that the two coordinate systems have the same origin.

Example

cos 2.5 m cos35x r 2.0 m

sin 2.5 m sin35y r 1.4 m

The x coordinate is found as

and the y coordinate

A certain corner of a room is selected as the origin of a rectangular coordinate system. If a fly is crawling on an adjacent wall at a point having coordinates (2.0, 1.0), where the units are meters, what is the distance of the fly from the corner of the room?

Example

2 22 2 = 2.0 m 1.0 m = d x y 2.2 m

The x distance out to the fly is 2.0 m and the y distance up to the fly is 1.0 m. Thus, we can use the Pythagorean theorem to find the distance from the origin to the fly as,

Express the location of the fly in example before in polar coordinates.

Example

1.0 mtan 0.5

2.0 m

y

x 1tan 0.50 27

2.2 m and 27r

The distance from the origin to the fly is r in polar coordinates, and this was found to be 2.2 m. The angle is the angle between r and the horizontal reference line (the x axis in this case). Thus, the angle can be found as

and

The polar coordinates are

Summary Equations are the tools of physics

Understand what the equations mean and how to use them

Carry through the algebra as far as possible Substitute numbers at the end

Be organized