Chapter 2 a Mathematical Toolkit

Mr. BoroskyPhysics Section 1.1 NotesPage 1 of 7

Chapter 1 A Physics Toolkit

In this chapter you will:

Use mathematical tools to measure and predict.
Apply accuracy and precision when measuring.
Display and evaluate data graphically.

Sections

Section 1.1: Mathematics and Physics

Section 1.2: Measurement

Section 1.3: Graphing Data

Section 1.1 Mathematics and Physics

Objectives

Demonstrate scientific methods.
Use the metric system.
Evaluate answers using dimensional analysis.
Perform arithmetic operations using scientific notation.

What is Physics?

Physics is a branch of science that involves the study of the physical world: energy, matter, and how they are related.

Learning physics will help you to understand the physical world.

The goal of this course is to help you understand the physical world.

You can use the problem solving skills you use in physics in many disciplines.

Physics uses mathematics as a powerful language.

Mathematics is the language of Physics.

In physics, equations are important tools for modeling observations and for making predictions.

Physicists rely on theories and experiments with numerical results to support their conclusions.

Example Problem 1 Electric Current

V = IR

120 = .75 R

160 Ohms = R

SI UNITS

The example problem uses different units of measurement to communicate the variables and the result. It is helpful to use units that everyone understands.

Scientific institutions have been created to define and regulate measures.

The worldwide scientific community and most countries currently use an adaptation of the metric system to state measurements.

Metric System – system of measurement that is based on powers of ten. It was created by French scientists in 1795.

The Système International d’Unités, or SI- uses seven base quantities, which are shown in the table below (Table 1-1 p. 5). (International System of Units)

SI – International System of Units. This is abbreviated SI because in Europe it is called Systeme International.

Almost every country except the US uses the Metric System in everyday life. The Scientific Community (including the US) uses an adaptation of the Metric System (SI) to make measurements.

Base Quantities (or Fundamental Units) – set of units on which a measurement system is based. They were originally defined in terms of direct measurements.

Second – standard unit of time.

The second was first defined as 1/86,400 of the mean solar day. Mean Solar Day is the average length of the day over a period of one year.

In 1967 the second was redefined in terms of the frequency of one type of radiation emitted by a Cesium-133 atom.

Meter – standard SI unit of length.

Meter was first defined as one-ten-millionth (10-7) of the distance from the North Pole to the equator measured along a line passing through Lyons, France.

In the 20th Century Physicists found that light could be used to make very precise measurements of distances.

In 1960, the meter was redefined as a multiple of a wavelength of light emitted by Krypton-86. By 1982, a more precise length measurement defined the meter as the distance light travels in 1/299,792,458 second in a vacuum.

Kilogram – standard SI unit of mass of an object.

Kilogram is the only SI unit not defined in terms of the properties of atoms. It is the mass of a Platinum-Iridium metal cylinder kept near Paris.

Derived Units – are created by combining the base units in various ways. The unit of a quantity that consists of combinations of fundamental or base units. A common derived unit is the meter per second (m/s), which is used to measure speed.

The SI system is regulated by the International Bureau of Weights and Measures in Sèvres, France.

This bureau and the National Institute of Science and Technology (NIST) in Gaithersburg, Maryland, keep the standards of length, time, and mass against which our meter sticks, clocks, and balances are calibrated.

The ease of switching between units is another feature of the metric system.

Prefixes – are used to change SI units by powers of 10.

To convert between SI units, multiply or divide by the appropriate power of 10.

Prefixes are used to change SI units by powers of 10, as shown in the table below.

See Table 1-2 for the Prefixes. Notice once we reach three then we use factors of 3 such as 6, 9, 12, etc.

To use the SI units effectively you need to know the meanings of the prefixes. Make sure you know the Prefixes listed in Table 1-2. And Deka and Hecto.

When using the prefixes we usually use powers of 1, 2 or factors of 3.

Note: 101 is deka with symbol da and 102 is hecto with symbol h

Extra Credit. Find the Prefixes for 104 , 105 , 10-4 , 10-5

DIMENSIONAL ANALYSIS

You often will need to use different versions of a formula, or use a string of formulas, to solve a physics problem.

To check that you have set up a problem correctly, write the equation or set of equations you plan to use with the appropriate units.

Dimensional Analysis - method of treating units as algebraic quantities, which can be cancelled. It can be used to check that an answer will be in the correct units. It is also used in choosing conversion factors.

Conversion Factor - is a multiplier equal to 1. For example, because 1 kg = 1000 g, you can construct the following conversion factors:

Choose a conversion factor that will make the units cancel, leaving the answer in the correct units.

SIGNIFICANT DIGITS

Significant Digits - the valid digits in a measurement.

Uncertain Digit - the last digit given for any measurement.

Rules for Significant Digits

1. All Non-Zero Digits are Significant.

2. Final Zeros after the Decimal Point are Significant.

3. Zeros between significant digits are Significant.

4. Zeros used only as placeholders are NOT Significant.

Note: #s such as 1000 are normally written in scientific notation so you can tell how many significant digits. 1 * 103 has 1, 1.0 * 103 has 2, 1.00 * 103 has 3, and 1.000 * 103 has 4

Also Counting #s and Conversion Factors are EXACT so they have INFINITE Significant Digits

When you perform any arithmetic operation, it is important to remember that the result never can be more precise than the least-precise measurement.

To add or subtract measurements, first perform the operation, then round off the result to correspond to the least-precise value involved.

To multiply or divide measurements, perform the calculation and then round to the same number of significant digits as the measurement with the least number of significant digits.

Note that significant digits are considered only when calculating with measurements.

SCIENTIFIC METHODS

Scientific Method – a systematic method of observing, experimenting, and analyzing to answer questions about the natural world.

Written, oral, and mathematical communication skills are vital to every scientist.

Hypothesis - an educated guess about how variables are related.

A hypothesis can be tested by conducting experiments, taking measurements, and identifying what variables are important and how they are related. Based on the test results, scientists establish models, laws, and theories.

Scientific Method Steps

1. State the Problem

2. Gather Information

3. Form a Hypothesis

4. Test the Hypothesis

5. Analyze Data

6. Draw Conclusions

Scientific Models - are based on experimentation.

Scientific Law - is a rule of nature that sums up related observations to describe a pattern in nature. A well established rule about the natural world that sums up, but does not explain a pattern in nature.

Scientific Theory - is an explanation based on many observations supported by experimental results. An explanation based on numerous observations, supported by experimental results, that may explain why things work the way they do.

Question 1

The potential energy, PE, of a body of mass, m, raised to a height, h, is expressed mathematically as PE = mgh,where g is the gravitational constant. If m is measured in kg, g in m/s2, h in m, and PE in joules, then what is 1 joule described in base unit?

PE = mgh

J = kg (m/s2)(m)

SoJ = kg*m2/s2

Question 2

A car is moving at a speed of 90 km/h. What is the speed of the car in m/s? (Hint: Use Dimensional Analysis)

90 km | 1 h | 1000 m = 25 m/s

h | 3600 s | 1 km

Question 3

Which of the following representations is correct when you solve 0.030 kg + 3333 g using scientific notation?

3.4×103 g
3.36×103 g
3×103 g
3363 g

.030 kg + 3.333 kg = 3.363 kg so 3.4 kg since 2 Sig. Dig.

Or 3.0 * 101 g + 3333 g = 3363 g so 3.4 * 103 g since 2 Sig. Dig.