General Physics (PHY 2130) Lecture 2 •  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

Problem Solving Strategy Given: θ = 39.0 angle: distance: d = 46.0m

Find: Height=? Fig. 1.7, p.14 Slide 13

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! height of building , dist . height = dist . × tan α = (tan 39.0 )(46.0 m) = 37.3 m tan α =

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

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

quantity

denotes the physical nature of a

  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 Area Volume Velocity (speed) Acceleration

  Example

L L2 L3 L/T L/T2

m (SI) m2 (SI) m3 (SI) m/s (SI) m/s2 (SI)

of dimensional analysis

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

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

[L] [length of the pendulum] [ period] = [T ] = = 2 [acceleration of gravity] [L / T ]

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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 1m = 39.37 in = 3.281 ft

1 ft = 0.3048 m = 30.48 cm 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? 100

km km 1 mile miles = 100 ⋅ ≈ 62 h h 1.609 km h

Example 3. The density of air. 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

3

kg ⎛ 1 slug ⎞⎛ 1 m ⎞ −3 3 ⎟⎟⎜ 1.3 3 ⎜⎜ = 2 . 5 × 10 slugs/ft ⎟ m ⎝ 14.59 kg ⎠⎝ 3.28 feet ⎠

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Prefixes ► Prefixes

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

Prefix Abbrev.

1015 109 106 103 10-2 10-3 10-6 10-9

peta giga mega kilo centi milli micro nano

P G M k P m µ n

Distance from Earth to nearest star Mean radius of Earth Length of a housefly Size of living cells Size of an atom

40 Pm 6 Mm 5 mm 10 µm 0.1 nm

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

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

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

m = 325 mg = 325 ×10−3 g = 0.325 g

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

17400 = 1.74 × 10 4

3 significant figures

17400. = 1.7400 × 10 4

5 significant figures

17400.0 = 1.74000 × 10

4

6 significant figures

Operations with Significant Figures ►  Accuracy Example:

-- number of significant figures

meter stick:

± 0.1 cm

►  When

multiplying or dividing, round the result to the same accuracy as the least accurate measurement 2 significant figures Example:

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

►  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

Order of Magnitude ►  Approximation

based on a number of assumptions

  may need to modify assumptions if more precise results are needed 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)!

►  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”

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.

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

Dependent variable here

Independent variable here 21

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.

The given data:

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

23

103 102.5

temp (F)

102 101.5 101 100.5 100 99.5 10

11

12

13

time (hours) 24

(a)

(b)

(c)

Reading from the graph: 101.8 °F.

T2 − T1 101.8 °F − 100.0 °F slope = = = 0.9 °F/hour t 2 − t1 12.0 hr − 10.0 hr

No.

25