Chapter 1- Basic Physics Tools and Errro Analysis Error.

Chapter 1- Basic Physics Tools and Errro Analysis

Error

Why Bother?

• The knowledge we have of the physical world is obtained by doing experiments and making measurements.

• It is important to understand how to express such data and how to analyze and draw meaningful conclusions from it.

Why Bother?

• ALL measurements of physical quantities are subject to uncertainties.

• It is never possible to measure anything exactly.

• in order to draw valid conclusions the error must be indicated and dealt with properly.

Example: Your Height is 5' 8“. How accurate is this?

The height of a person depends on :• how straight she stands, • Did she just got up from lying horizontally • Did she has her shoes on• How her hair is made up. • A quantity such as height is not exactly

defined without specifying many other circumstances.

That’s Not All…..

• Even if you could precisely specify the "circumstances," your result would still have an error associated with it.

• The scale you are using is of limited accuracy • when you read the scale, you may have to

estimate a fraction between the marks on the scale, etc.

The two essential components of a physical measurement

• (1) A numerical value (in a specified system of units) giving the best estimate possible of the quantity measured

• (2) the degree of uncertainty associated with this estimated value.

• For example, a measurement of the width of a table would yield a result such as

• 95.3cm +/- 0.1 cm.

Significant Figures

• Definition: The significant figures of a quantity are the meaningful digits in it.

• 1. Nonzero digits are always significant. • 2.All final zeros after the decimal point are significant• 3.Zeros between two other significant digits are

always significant• 4. Zeros used solely for spacing the decimal point are

not significant.

Significant Figures

• 1. Any digit that is not zero is significant.

549 1.892

Significant Figures

• 2. Zeros between non zero digits are significant.

• 4023• 68907• 101

Significant Figures

• 3. Zeros to the left of the first non zero digit are not significant

• 0.000034 = 3.4x10-5 • 0.01 = 1x10-2 • 0.00416 = 4.16x10-3

Significant Figures

• For numbers with decimal points, zeros to the right of a non zero digit are significant.

• 2.00 has three significant figures • 0.050 has two significant figures. • For this reason it is important to keep the trailing

zeros to indicate the actual number of significant figures.

How many Significant Figures does each number have?

• 1023.00• 2208• 0.000056• 3,000• 5550• 609.000• 8.000001• 0.002

Percent Error• To express the magnitude of the error (or deviation)

between two measurements scientists invariably use percent error .

• Example: The length of a box is measured to be 12.5cm, and the actual value is 12.0 cm. What is the percent error in the measurement?

• % Error = 12.5cm-12.0cm x 100 = 0.5 x 100 = 4.16%• 12.0cm 12.0

Vocabulary• 1. Physics – the study of the physical world: energy,

matter and how they are related.• 2. Dimensional Analysis – The method of treating

the units as algebraic quantities which can be cancelled.

• Significant Figures – The valid digits in a measurement

• Scientific Method – • Hypothesis – An educated guess about how

variables are related.• Scientific Law – A rule of nature that sums up

related observations to describe a pattern in nature.

Vocabulary• Scientific Theory – An explanation based on many

observations supported by experimental results.• Measurement – A comparison between an

unknown quantity and a standard.• Precision – The degree of exactness.• Accuracy – How well the results of a

measurement agree with the “real” value.• Parallax – The apparent shift in the position of an

object when it is viewed from different angles.

Vocabulary• Independent Variable – The factor that is

changed or manipulated during the experiment.

• Dependent variable – the factor that depends on the independent variable.

• Line of Best Fit – A line drawn as close as possible to all the data points .

Example: when you attach different masses to a spring to see how much it gets stretched, Which is the Independent Variable? Dependent Variable?

How to Plot Graphs• 1. Identify the independent variable and dependent

variables in your data.• 2. The independent variable is plotted on the x-axis.• 3. The dependent variable is plotted on the y-axis.• 4. Determine the ranges of both variables to be plotted.• 5. Decide if (0,0) is a valid data point.• 6. Spread the data out as much as possible. Use convenient

divisions such as 2, 5, 10.• 7. Number and label the horizontal axis. Include the units,

e.g. Mass (grams)• 8. Draw a Best Fit Line/Curve• 9. Give the graph a title that clearly tells what the graph

represents.

How to Draw a Line of Best Fit• Average all the x measurements• Average all the y measurements• Plot this point (xavg, yavg) on the graph. This will

be your pivot point. • Use the pivot point to draw a line that passes

as much data points as possible and has about equal numbers of data points on either side of the line.

Linear Relationships• When the best-fit line is a straight line, the

relationship between the independent variable and dependent variable is linear.

• The equation of the line is y = mx + b, where • m = slope of the line = Rise = y• Run x• b = y-intercept

Linear Relationships

• 1. Temperature – Celsius and Farenheit • C = (5/9) (F-32)• 2. Exchange Rates – E = 0.7749D Euro to dollar• 3. Cell phone costs/month C = 0.05m + 20• 4. Distance travelled D = 60t

Nonlinear Relationships

• When the graph is not a straight line, the relationship between the independent and the dependent variable is NOT linear.

• Common nonlinear graphs: • Quadratic: y = ax2 + bx + c• Inverse: y = a/x

Quadratic Relationship

Real applications of quadratic equations

• 1. Path of planetary motion• 2. Stopping distance of a braking car.• 3. Trajectory of a ball that is thrown• -- artillery calculations• 4.Bernoulli’s Principle--predict the behaviour

of the flow of air over the wing of an aircraft and to see why an aircraft flies.

Inverse Relationship

• 1. Pressure and volume of a gas• 2. Price and demand in economics If the local Starbucks lowers their price of a tall coffee from $1.75 to $1.65, the quantity demanded will rise from 45 coffees an hour to 48 coffees an hour.3. Mortality Rates And Performance In The Hospital Quality Alliance Measures

Order of Magnitude

• Used to make a rough comparison between compare numbers.

• Order of Magnitude of 1 = 101

• Order of Magnitude of 2 = 102

• Order of Magnitude of 3 = 103 etc.

How to Find the Order of Magnitude of a number

• Write the number in Scientific Notation• If the mantissa (left side) is greater than 5, then go up one

more power. • Example: 8.9 x 104

It is greater than 5.0x104 Therefore 8.9x 104 would have an Order of Magnitude of 5.

• **check: 89,000 is closer to 100,000 • than 10,000.

Order of Magnitude Number Scientific

NotationGreater than 5 In the mantissa

Order of Magnitude

2789 2.789x103 No 3

5510 5.510 x 103 Yes 4

97000 9.7000 x 104

0.00678 6.78 x 10-3

0.00456 4.56 x 10-3