Chapter 1: A Physics Toolkit PHYSICS Principles and Problems

Chapter 1: A Physics ToolkitChapter 1: A Physics Toolkit

PHYSICS Principles and

Problems

BIG IDEA

Physicists use scientific methods to investigate energy and matter.

CHAPTER

1 A Physics Toolkit

Section 1.1 Methods of Science

Section 1.2 Mathematics and Physics

Section 1.3 Measurement

Section 1.4 Graphing Data

CHAPTER

1 Table Of Contents

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Essential Questions

• What are the characteristics of scientific methods?

• Why do scientists use models?

• What is the difference between a scientific theory and a scientific law?

• What are some limitations of science?

MAIN IDEA

Scientific investigations do not always proceed with identical steps but do contain similar methods.

SECTION1.1

Methods of Science

New Vocabulary

• Physics

• Scientific methods

• Hypothesis

Review Vocabulary

• Control the standard by which test results in an experiment can be compared.

• Model

• Scientific theory

• Scientific law

SECTION1.1

Methods of Science

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.

What is Physics?

SECTION1.1

Methods of Science

• Although physicists do not always follow a rigid set of steps, investigations often follow similar patterns called scientific methods.

• Depending on the particular investigation, a scientist might add new steps, repeat some steps or skip steps altogether.

Scientific Methods

SECTION1.1

Methods of Science

• Many investigations begin when someone observes an event in nature and wonders why or how it occurs.

• The question of “why” or “how” is the problem.

– Many questions have been asked throughout history: why objects fall to Earth, what causes day and night, how to generate electricity…

– Often the investigation into one problem may lead to more questions and more investigations.

Scientific Methods (cont.)

SECTION1.1

Methods of Science

• Researching already known information about a problem, helps to fine-tune the question and form it into a hypothesis.– Hypothesis is a possible explanation for a problem

using what you know and have observed.


SECTION1.1

Methods of Science

• Hypotheses can be tested by different means:

– Observations

– Models

– Experiments

• Test the effect of one thing on another, using a control.


SECTION1.1

Methods of Science

• An important part of every investigation includes recording observations and organizing data into easy-to-read tables and graphs.

• Based on the analysis of the data, the next step is to decide whether the hypothesis is supported.

– If supported, the data must be reproducible many times.

– If not supported, the hypothesis must be reconsidered.


SECTION1.1

Methods of Science

• Sometimes, scientists cannot see everything they are testing. They might be observing an object that is too large or too small, a process that takes too much time to see completely, or a material that is hazardous.

• A model is a representation of an idea, event, structure, or object that helps people better understand it.

Models

SECTION1.1

Methods of Science

• A scientific theory is an explanation of things or events based on knowledge gained from many observations and investigations.

– This is not a hypothesis, this is what a hypothesis becomes after numerous trials of data supporting the hypothesis.

– A theory is never permanent, it can change as new data and information becomes available.

Scientific Theories and Laws

SECTION1.1

Methods of Science

• A scientific law is a statement about what happens in nature and seems to be true all the time.

– Laws tell you what will happen under certain conditions, but they do not explain why or how something happens.

Ex. Gravity

• The law of gravity states that any one mass will attract another mass.

• There are many theories proposed to explain how the law of gravity works.

Scientific Theories and Laws (cont.)

SECTION1.1

Methods of Science

• Science cannot explain or solve every question.

• A scientific question must be testable and verifiable.

– Questions about opinions, values or emotions are not scientific because they cannot be tested.

The Limitations of Science

SECTION1.1

Methods of Science

The similar patterns used, by all branches of science, in an investigation are called?

A. Hypotheses

B. Scientific theories

C. Scientific methods

D. Scientific laws

SECTION1.1

Section Check

“In a closed-system, mass is always conserved” is an example of which of the following?

A. Scientific law

B. Scientific theory

C. Hypothesis

D. Model

SECTION1.1

Section Check

MAIN IDEA

We use math to express concepts in physics.

Essential Questions

• Why do scientists use the metric system?

• How can dimensional analysis help evaluate answers?

• What are significant figures?

SECTION1.2

Mathematics and Physics

Review Vocabulary

• SI International System of Units – the improved, universally accepted version of the metric system that is based on multiples of ten.

New Vocabulary

• Dimensional analysis

• Significant figures

SECTION1.2


• Physicists often use the language of mathematics.

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

Mathematics in Physics

SECTION1.2


• In order to communicate results that everyone can understand, the worldwide scientific community uses an adaptation of the metric system called Systeme International d’Unites or SI.

• The SI system of measurement uses seven base quantities.

SI Units

SECTION1.2


• The base quantities were originally defined in terms of direct measurements. Other units, called derived units, are created by combining the base units in various ways.

• 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 metersticks, clocks, and balances are calibrated.

SI Units (cont.)

SECTION1.2


• Another feature in the SI system is the ease of converting units.

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

• Prefixes are used to change SI base units to powers of 10.

SECTION1.2


SI Units (cont.)

Dimensional Analysis

• You will often 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.

SECTION1.2


• Before performing calculations, check that the answer will be in the expected units.

• For example, if you are finding a speed and you see that your answer will be measured in s/m, you know you have made an error in setting up the problem.

• This method of treating the units as algebraic quantities, which can be cancelled, is called dimensional analysis.

Dimensional Analysis (cont.)

SECTION1.2


• Dimensional analysis is also used in choosing conversion factors.

• A conversion factor is a multiplier equal to 1. For example, because 1 kg = 1000 g, you can construct the following conversion factors:

SECTION1.2



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

• For example, to convert 1.34 kg of iron ore to grams, do as shown below:

SECTION1.2



• A meterstick is used to measure a pen and you find the end of the pen is in between 138 and 139mm. You estimate that the pen is two-tenths of a millimeter past the 138 mark and record the measurement as 138.2mm.

• This measurement has four valid digits: three you are sure of, and one you estimated.

• The valid digits in a measurement are called significant figures.

• However, the last digit given for any measurement is the uncertain digit.

SECTION1.2


Significant Figures

Significant Figures (cont.)

• All nonzero digits in a measurement are significant, but not all zeros are significant.

• Consider a measurement such as 0.0860 m. Here the first two zeros serve only to locate the decimal point and are not significant.

• The last zero, however, is the estimated digit and is significant.

SECTION1.2


• When you perform any arithmetic operation, it is important to remember that the result can never 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. Ex. 3.86m + 2.4m = 6.3m

SECTION1.2



• To multiply or divide measurements:

–Perform the calculation and then round to the same number of significant digits as the least-precise measurement. Ex. 409.2km/11.4L = 35.9km/L

• Note: Significant digits are considered only when calculating with measurements. There is no uncertainty associated with counting (4 washers) or exact conversion factors (24 hours in 1 day).

SECTION1.2



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 units?

SECTION1.2

Section Check

A. 1 kg·m/s

B. 1 kg·m/s2

C. 1 kg·m2/s

D. 1 kg·m2/s2

Answer

Reason:

SECTION1.2

Section Check

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)

SECTION1.2

Section Check

A. 2.5×101 m/s

B. 1.5×103 m/s

C. 2.5 m/s

D. 1.5×102 m/s

Answer

Reason:

SECTION1.2

Section Check

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

SECTION1.2

Section Check

A. 3.4×103 g

B. 3.36×103 g

C. 3×103 g

D. 3.363×103 g

Answer

Reason: 0.030 kg is the same as 30g and can be written as 3.0 101 g which has 2 significant digits, the number 3 and the zero after 3.

3333 has four significant digits; all four threes.

Therefore, our answer should contain only 2 significant digits.

SECTION1.2

Section Check

MAIN IDEA

Making careful measurements allows scientists to repeat experiments and compare results.

Essential Questions

• Why are the results of measurements often reported with an uncertainty?

• What is the difference between precision and accuracy?

• What is a common source of error when making a measurement?

SECTION1.3

Measurement

Review Vocabulary

• Parallax the apparent shift in the position of an object when it is viewed from different angles.

New Vocabulary

• Measurement

• Precision

• Accuracy

SECTION1.3

Measurement

• A measurement is a comparison between an unknown quantity and a standard.

–Ex. Measuring the mass of a rolling cart. The unknown quantity is the cart, the standard is the gram as defined the instrument being used.

• Measurements quantify observations.

• Careful measurements enable you to derive the relationship between any two quantities.

What is a Measurement?

SECTION1.3

Measurement

• When a measurement is made, the results are often reported with uncertainty.

• Therefore, before fully accepting new data, other scientists examine the experiment, looking for possible sources of errors, and try to reproduce the results.

• A new measurement that is within the margin of uncertainty confirms the old measurement.

Comparing Results

SECTION1.3

Measurement

Precision Versus Accuracy

Click image to view the movie.

SECTION1.3

Measurement

• To assure precision and accuracy, instruments used to make measurements need to be used correctly.

• This is important because one common source of error comes from the angle at which an instrument is read.

Techniques of Good Measurement

SECTION1.3

Measurement

• Scales should be read with one’s eye straight in front of the measure.

• If the scale is read from an angle, as shown in figure (b), you will get a different, and less accurate, value.

Techniques of Good Measurement

(a)

(b)

• The difference in readings is caused by parallax, which isthe apparent shift in the position of an object when it is viewed from different angles.

SECTION1.3

Measurement

Ronald, Kevin, and Paul perform an experiment to determine the value of acceleration due to gravity on Earth (which most scientists agree is about 980 cm/s2). The following results were obtained: Ronald — 961 ± 12 cm/s2, Kevin — 953 ± 8 cm/s2, and Paul — 942 ± 4 cm/s2. Determine who has the most accurate and precise value.

A. Kevin got the most precise and accurate value.

B. Ronald’s value is the most accurate, while Kevin’s value is the most precise.

C. Ronald’s value is the most accurate, while Paul’s value is the most precise.

D. Paul’s value is the most accurate, while Ronald’s value is the most precise.

SECTION1.3

Measurement

Answer

Reason: Ronald’s answer is closest to 980 cm/s2. Hence, Ronald’s result is the most accurate. However, Paul’s error is only ±4 cm/s2. Hence, Paul’s result is the most precise.

SECTION1.3

Section Check

What is the precision of an instrument?

A. the smallest divisions marked on the instrument

B. the least count written on the instrument

C. one-half the least count written on the instrument

D. one-half the smallest division written on the instrument

SECTION1.3

Section Check

Answer

Reason: Precision depends on the instrument and the technique used to make the measurement. Generally, the device with the finest division on its scale produces the most precise measurement. The precision of a measurement is one-half of the smallest division of the instrument.

SECTION1.3

Section Check

A 100-cm long rope was measured with three different measuring tapes. The answer obtained with the three measuring tapes were: 1st measuring tape — 99 ± 0.5 cm, 2nd measuring tape — 98 ± 0.25 cm, and 3rd measuring tape — 99 ± 1 cm. Which measuring tape is the most precise?

A. 1st measuring tape

B. 2nd measuring tape

C. 3rd measuring tape

D. Both measuring tapes 1 and 3

SECTION1.3

Section Check

Answer

Reason: Precision depends on the instrument. The 2nd measuring tape has an error of only ±0.25 cm and is therefore the most precise.

SECTION1.3

Section Check

MAIN IDEA

Graphs make it easier to interpret data, identify trends and show relationships among a set of variables.

Essential Questions

• What can be learned from graphs?

• What are some common relationships in graphs?

• How do scientists make predictions?

SECTION1.4

Graphing Data

Review Vocabulary

• Slope on a graph, the ratio of vertical change to horizontal change.

New Vocabulary

• Independent variable

• Dependent variable

• Line of best fit

• Linear relationship

• Quadratic relationship

• Inverse relationship

SECTION1.4

Graphing Data

• A variable is any factor that might affect the behavior of an experimental setup.

• The independent variable is the factor that is changed or manipulated during the experiment.

• The dependent variable is the factor that depends on the independent variable.

Identifying Variables

SECTION1.4

Graphing Data

Click image to view the movie.

Identifying Variables (cont.)

SECTION1.4

Graphing Data

• Scatter plots of data may take many different shapes, suggesting different relationships.

• Three of the most common relationships include linear relationships, quadratic relationships and inverse relationships.

Linear Relationships

SECTION1.4

Graphing Data

• When the line of best fit is a straight line, as in the figure, the dependent variable varies linearly with the independent variable. This relationship between the two variables is called a linear relationship.

• The relationship can be written as an equation.


SECTION1.4

Graphing Data

• The slope is the ratio of the vertical change to the horizontal change. To find the slope, select two points, A and B, far apart on the line. The vertical change, or rise, Δy, is the difference between the vertical values of A and B. The horizontal change, or run, Δx, is the difference between the horizontal values of A and B.


SECTION1.4

Graphing Data

• As presented in the previous slide, the slope of a line is equal to the rise divided by the run, which also can be expressed as the change in y divided by the change in x.

• If y gets smaller as x gets larger, then Δy/Δx is negative, and the line slopes downward.

• The y-intercept, b, is the point at which the line crosses the y-axis, and it is the y-value when the value of x is zero.


SECTION1.4

Graphing Data

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

• There are many types of nonlinear relationships in science. Two of the most common are the quadratic and inverse relationships.

Nonlinear Relationships

SECTION1.4

Graphing Data

• The graph shown in the figure is a quadratic relationship.

• A quadratic relationship exists when one variable depends on the square of another.


SECTION1.4

Graphing Data


• A quadratic relationship can be represented by the following equation:

SECTION1.4

Graphing Data

• The graph in the figure shows how the current in an electric circuit varies as the resistance is increased. This is an example of an inverse relationship.

• In an inverse relationship, a hyperbola results when one variable depends on the inverse of the other.


SECTION1.4

Graphing Data


• An inverse relationship can be represented by the following equation:

SECTION1.4

Graphing Data

• There are various mathematical models available apart from the three relationships you have learned. Examples include sinusoids, which are used to model cyclical phenomena, and exponential decay curves, which are used to model radioactivity.

• Combinations of different mathematical models represent even more complex phenomena.


SECTION1.4

Graphing Data

• Relations, either learned as formulas or developed from graphs, can be used to predict values you have not measured directly.

• Physicists use models to accurately predict how systems will behave: what circumstances might lead to a solar flare, how changes to a circuit will change the performance of a device, or how electromagnetic fields will affect a medical instrument.

Predicting Values

SECTION1.4

Graphing Data

Which type of relationship is shown by the following graph?

A. Linear

B. Inverse

C. Parabolic

D. Quadratic

SECTION1.4

Section Check

Answer

Reason: In an inverse relationship, a hyperbola results when one variable depends on the inverse of the other.

SECTION1.4

Section Check

What is a line of best fit?

A. the line joining the first and last data points in a graph

B. the line joining the two center-most data points in a graph

C. the line drawn as close to all the data points as possible

D. the line joining the maximum data points in a graph

SECTION1.4

Section Check

Answer

Reason: The line drawn closest to all data points as possible is called the line of best fit. The line of best fit is a better model for predictions than any one or two points that help to determine the line.

SECTION1.4

Section Check

Which relationship can be written as y = mx + b?

A. Linear relationship

B. Quadratic relationship

C. Parabolic relationship

D. Inverse relationship

SECTION1.4

Section Check

Answer

Reason: A linear relationship can be written as y = mx + b, where m is the slope and b is the y-intercept.

SECTION1.4

Section Check

Physics Online

Study Guide

Chapter Assessment Questions

Standardized Test Practice

A Physics ToolkitCHAPTER

1

Resources

• Scientific methods include making observations and asking questions about the natural world.

• Scientists use models to represent things that may be too small or too large, processes that take too much time to see completely, or a material that is hazardous.

Methods of ScienceSECTION1.1

Study Guide

• A scientific theory is an explanation of things or events based on knowledge gained from observations and investigations. A scientific law is a statement about what happens in nature, which seems to be true all the time.

• Science can not explain or solve everything. Questions about opinions or values can not be tested.

Methods of ScienceSECTION1.1

Study Guide

• Using the metric system helps scientists around the world communicate more easily.

• Dimensional analysis is used to check that an answer will be in the correct units.

• Significant figures are the valid digits in a measurement.

Mathematics and PhysicsSECTION1.2

Study Guide

• Measurements are reported with uncertainty because a new measurement that is within the margin of uncertainty confirms the old measurement.

• Precision is the degree of exactness with which a quantity is measured. Accuracy is the extent to which a measurement matches the true value.

• A common source of error that occurs when making a measurement is the angle at which an instrument is read. If the scale of an instrument is read an angle, as opposed to eye level, the measurement will be less

accurate.

MeasurementSECTION1.3

Study Guide

• Graphs contain information about the relationships among variables. Patterns that are not immediately evident in a list of numbers are seen more easily when the data are graphed.

Graphing DataSECTION1.4

Study Guide

• Common relationships shown in graphs include linear relationships, quadratic relationships and inverse relationships. In a linear relationship, the dependent variable varies linearly with the independent variable. A quadratic relationship occurs when one variable depends on the square of an another. In an inverse relationship, one variable depends on the inverse of the other variable.

• Scientists use models and relationships between variables to make predictions.

Graphing DataSECTION1.4

Study Guide

How will you express 1 nm in m?

A. 1×10-3 m

B. 1×10-6 m

C. 1×10-9 m

D. 1×10-1 m

Chapter Assessment


1

Reason: 1 nm is read as 1 nanometer. The prefix nano stands for 10-9.

Chapter Assessment


1

Add the following numbers and write the answer using the proper number of significant digits: 12.3 + 1.2 + 123.

A. 136.5

B. 1.4 × 102

C. 137

D. 1.37 × 102

Chapter Assessment


1

Reason: The last digit in 12.3 and 1.2 are both in the tenth’s place. However, the last digit in 123 is in the one’s place. Therefore, the last digit of the answer should be in the one’s place.

Chapter Assessment


1

Rewrite 3.650 with only 2 significant digits.

A. 3.7

B. 3.6

C. 3.65

D. 0.360 × 101

Chapter Assessment


1

Reason: The last reported digit would be the 6. The digit to the right is a 5 followed by a zero. Therefore since the 6 is even it remains so and the answer would be 3.6.

Chapter Assessment


1

If 15 different individuals perform an experiment, and 15 answers are obtained, which answer will be accepted as the most accurate?

A. the answer obtained by the highest number of persons

B. the eighth number if all the numbers are arranged in an ascending order

C. the answer nearest to the expected answer

D. the average of all 15 answers

Chapter Assessment


1

Reason: Accuracy describes how well the result of a measurement agrees with the expected value.

Chapter Assessment


1

A quadratic relationship between two variables is written as ____.

A.

B.

C.

D.

Chapter Assessment


1

Reason: A quadratic relationship between two variables is written asy = ax2 + bx + c.

Chapter Assessment


1

Two laboratories use radiocarbon dating to measure the age of two wooden spear handles found in the same grave. Lab A finds an age of 2250 40 years for the first object; lab B finds an age of 2215 50 years for the second object. Which of the following is true?

A. Lab A’s reading is more accurate than lab B’s.

B. Lab A’s reading is less accurate than lab B’s.

C. Lab A’s reading is more precise than lab B’s.

D. Lab A’s reading is less precise than lab B’s.



1

Which of the following is equal to 86.2 cm?

A. 8.62 m

B. 0.862 mm

C. 8.62×10-4 km

D. 862 dm



1

Jario has a homework problem to do involving time, distance, and velocity, but he has forgotten the formula. The question asks him for a measurement in seconds, and the numbers that are given have units of m/s and km. What could Jario do to get the answer in seconds?

A. Multiply the km by the m/s, then multiply by 1000.

B. Divide the km by the m/s, then multiply by 1000.

C. Divide the km by the m/s, then divide by 1000.

D. Multiply the km by the m/s, then divide by 1000.



1

What is the slope of the graph?

A. 0.25 m/s2

B. 0.4 m/s2

C. 2.5 m/s2

D. 4.0 m/s2



1

Which formula is equivalent to

A.

B.

C.

D.



1

Skip Around if You Can

You may want to skip over difficult questions and come back to them later, after you’ve answered the easier questions. This will guarantee more points toward your final score. In fact, other questions may help you answer the ones you skipped. Just be sure you fill in the correct ovals on your answer sheet.

Test-Taking Tip



1

Length of a Spring for Different Masses (1)


1

Chapter Resources

Length of a Spring for Different Masses (2)


1

Chapter Resources

Graph Indicating a Quadratic, or Parabolic, Relationship


1

Chapter Resources

Graph Showing the Inverse Relationship Between Resistance and Current


1

Chapter Resources

Documents

Chapter 1: A Physics Toolkit PHYSICS Principles and Problems