Lecture 2

General Physics (PHY 2130)

•  Significant figures •  Units •  Graphs

http://www.physics.wayne.edu/~apetrov/PHY2130/

Chapter 1

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Lightning Review

Last lecture: 1.  Introduction to physics and relevant math

  Scientific notation, percentages, etc.   Units: meter, kilogram, second (definitions)

Review Problem: A firefighter attempts to measure the height of the building by walking out a distance of 46.0 m from its base and shining a flashlight beam towards its top. He finds that when the beam is elevated at an angle of 39.0°, the beam just strikes the top of the building. Find the height of the building.

Problem Solving Strategy

Slide 13

Fig. 1.7, p.14

Given: angle: distance: Find: Height=?

mmdistheightdistbuildingofheight

3.37)0.46)(0.39(tantan.

,.

tan

==×=

=

α

α

Key idea: beam of light, building wall and distance from the building to the firefighter form a right triangle! Know: angle and one side, need to determine another side. NOTE: tangent is defined via two sides!

θ = 39.0

d = 46.0m

Evaluate answer: 1. Makes sense (a 37 m building is Ok) 2. Units are correct.

Problem Solving Strategy

1. Dimensions and Dimensional Analysis

► Dimensions are basic types of quantities that can be measured or computed. Examples are length, time, mass, electric current, and temperature.

► A unit is a standard amount of a dimensional quantity. There is a need for a system of units. SI units will be used throughout this class.

Dimensions

► Dimension denotes the physical nature of a quantity   dimension of some quantity, say, Q is denoted [Q]

► Dimensional analysis is a technique to check the correctness of an equation

► Dimensions (length, mass, time, combinations) can be treated as algebraic quantities   add, subtract, multiply, divide   quantities added/subtracted only if have same units

► Both sides of equation must have the same dimensions

Dimensions

►  Dimensions for commonly used quantities Length L m (SI) Area L2 m2 (SI) Volume L3 m3 (SI) Velocity (speed) L/T m/s (SI) Acceleration L/T2 m/s2 (SI)

  Example of dimensional analysis

distance = velocity · time L = (L/T) · T

9

Example: Use dimensional analysis to determine how the period of a pendulum depends on mass, the length of the pendulum, and the acceleration due to gravity (here the units are distance/time2).

Mass of the pendulum [M]

Length of the pendulum [L]

Acceleration of gravity [L/T2]

The period of a pendulum is how long it takes to complete 1 swing; the dimensions are time [T].

Solution: since the right dimension for the period is [T], we can get it from the quantities above as

[ period]= [T ]= [L][L /T 2 ]

=[length of the pendulum][acceleration of gravity]

Derived unit

►  A derived unit is composed of combinations of base units.

Example: The SI unit of energy is the joule.

1 joule = 1 kg m2/sec2

Derived unit Base units

2. 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 out

1 mile = 1609 m = 1.609 km 1 ft = 0.3048 m = 30.48 cm 1m = 39.37 in = 3.281 ft 1 in = 0.0254 m = 2.54 cm

Example 1. Scotch tape:

Example 2. Trip to Canada: Legal freeway speed limit in Canada is 100 km/h.

What is it in miles/h?

hmiles

kmmile

hkm

hkm 62

609.11100100 ≈⋅=

13

The density of air is 1.3 kg/m3. Change the units to slugs/ft3.

1 slug = 14.59 kg

1 m = 3.28 feet

333

3 slugs/ft 105.2feet 3.28

m 1kg 14.59

slug 1mkg 3.1 −×=⎟

⎠

⎞⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

Example 3. The density of air.

Prefixes

► Prefixes correspond to powers of 10 ► Each prefix has a specific name/abbreviation

Power Prefix Abbrev. 1015 peta P 109 giga G 106 mega M 103 kilo k 10-2 centi P 10-3 milli m 10-6 micro µ 10-9 nano n

Distance from Earth to nearest star 40 Pm Mean radius of Earth 6 Mm Length of a housefly 5 mm Size of living cells 10 µm Size of an atom 0.1 nm

Example: An aspirin tablet contains 325 mg of acetylsalicylic acid. Express this mass in grams.

Solution:

3325 325 10 0.325m mg g g−= = × =

Given: m = 325 mg Find: m (grams)=?

Recall that prefix “milli” implies 10-3, so

4. 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

4

4

4

1074000.10.17400107400.1.17400

1074.117400

×=

×=

×= 3 significant figures 5 significant figures 6 significant figures

Operations with Significant Figures

► Accuracy -- number of significant figures ► When multiplying or dividing, round the result

to the same accuracy as the least accurate measurement

► When adding or subtracting, round the result to

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

Example: 135 m + 6.213 m = 141 m

meter stick: cm1.0±

rectangular plate: 4.5 cm by 7.3 cm area: 32.85 cm2 33 cm2

2 significant figures

Example:

Example:

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

Example: John has 3 apples, Jane has 5 apples. Their numbers of apples are “of the same order of magnitude”

Question: McDonald’s sells about 250 million packages of fries every year. Placed back-to-back, how far would the fries reach? Solution: There are approximately 30 fries/package, thus: (30 fries/package)(250 . 106 packages)(3 in./fry) ~ 2 . 1010 in ~ 5 . 108 m,

which is greater then Earth-Moon distance (4 . 108 m)!

Important!

►  Order-of-magnitude estimates can be helpful in determining whether the answer you compute for a problem is reasonable.

Example: If you are asked to calculate the weight of a car, and come up with an answer of 10 lbs, you should re-check your calculation.

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Graphs

Experimenters vary a quantity (the independent variable) and measure another quantity (the dependent variable).

Dependent variable here

Independent variable here

22

Be sure to label the axes with both the quantity and its unit. For example:

Position (meters)

Time (seconds)

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Example: A nurse recorded the values shown in the table for a patient’s temperature. Plot a graph of temperature versus time and find (a) the patient’s temperature at noon, (b) the slope of the graph, and (c) if you would expect the graph to follow the same trend over the next 12 hours? Explain.

Time Decimal time Temp (°F)

10:00 AM 10.0 100.00 10:30 AM 10.5 100.45 11:00 AM 11.0 100.90

11:30 AM 11.5 101.35

12:45 PM 12.75 102.48

The given data:

24

99.5

100

100.5

101

101.5

102

102.5

103

10 11 12 13

time (hours)

tem

p (F

)

25

(a)

(b)

(c)

Reading from the graph: 101.8 °F.

F/hour 9.0hr 0.10hr 0.12

F 0.100F 8.101slope12

12 °=−

°−°=

−

−=

ttTT

No.