A Physics Toolkit

A Physics Toolkit

Chapter

1

A Physics Toolkit

Use mathematical tools to measure and predict.

Apply accuracy and precision when measuring.

Display and evaluate data graphically.

Chapter

1

In this chapter you will:

Mathematics and Physics

Be able to answer the question, what is physics?

Review the algebra required for this class.

Learn the GUESS problem solving strategy.

Use and do conversions with SI units.

Evaluate answers using dimensional analysis.

Perform arithmetic operations using scientific notation.

Learn about and use significant figures.

In this section you will:

Section

1.1


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

What is Physics?

Section

1.1a

Learning physics will help you to understand the physical world.

Physics is considered the basis for all other sciences:

- Biology, Chemistry, Astronomy, Geology, etc.

Physics is the fundamental science.


Many research physicists work in environments where they perform basic research in industry, research universities, and astronomical observations.

Physicists who find new ways to use physics are often employed by engineering, business, law, and consulting firms.

Physicists are also extremely valuable in areas such as computer science, medicine, communications, and publishing.

Finally, many physicists who love to see young people get excited about physics become teachers.

What does one do as a physicist?

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1.1


Every job has some relation to physics!

Athletes

- The laws of motion to lift, throw, push, hit, tackle, run, drag, jump, and crawl.

- The more an athlete and coach understand and use their knowledge of physics in their sport, the better the athlete will become.

What jobs do non-physicists hold that use physics every day?

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1.1


Automotive mechanics

◦ areas such as optics, electricity and magnetism, thermodynamics, and mechanics play greater roles

as vehicles become more and more complex.

CAT scan, computed tomography, and magnetic resonance imaging (MRI)

◦ technicians in hospitals and medical clinics who use such technology must have an understanding of what X-rays and magnetic resonance imaging are, how they behave, and how such high-tech instruments are to be used.

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Mathematics is the language of science.

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

Mathematics in Physics

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Simply: 2 + 5 x

√ Correct: 2 + 5 x

X Incorrect: ( 2 + 5 ) x = 7 x

Order Of Operations

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P arentheses and Brackets

E xponents

M ultiplication and Division (from left to right)

A ddition and Subtraction (from left to right)

Order Of Operations : PEMA

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Step 1: parentheses

Step 2: exponent

Step 3: multiplication

Step 4: addition

Step 5: solution

Order Of Operations : Example 1

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1.1

Step 1:

Step 2: distribute 8 into (x + 1)

Step 3: remove 1st parenthesis

Step 4: combine like terms

Step 5: within parenthesis, left to right, first comes division

Step 6: then multiplication

Step 7: simplify exponent

Step 8: solution

Order Of Operations : Example 2Section

1.1


Algebra Review with Physics

Section

1.1

Variables are used to represent concepts.

ex: d for displacement, t for time, p for momentum

Units also are abbreviated.

ex: m for meters, s2 for seconds squared

Do not confuse variables with units.

Subscripts are used to give more information about a variable.

ex: vave : average velocity

vi : initial velocity


Problem Solving Strategy

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1.1

Given: Write down given or known quantities. Draw a picture.

Unknown: Write down the unknown variable.

Equation: Find an applicable equation. Isolate the unknown variable.

Substitute numbers with units into the equation. Solve.

Sense: Are the units correct? Does the answer make sense?


Example: Electric Current

The voltage across a circuit equals the current multiplied by the resistance in the circuit. That is, V (volts) = I (amperes) × R (ohms). What is the resistance of a light bulb that has a 0.75 amperes current when plugged into a 120-volt outlet?

Section

1.1

Given:

I = 0.75 amperes

V = 120 volts

Unknown:

R = ?

Step 1: Given - Write the given quantities with units. Step 2: Unknown - Identify unknown variable.

Electric Current

Step 3: Equation. Rewrite the equation so that the unknown value is alone on the left.

Mathematics and PhysicsSection

1.1

V = IR

IR = V Reflexive property of equality.

Divide both sides by I.

Step 4: Substitute 120 volts for V, 0.75 amperes for I. Solve.

Resistance will be measured in ohms.R = 160

amperes 0.75volts 120R


Are the units correct?

1 volt = 1 ampere-ohm, so the answer in volts/ampere is in ohms, as expected.

Does the answer make sense?

120 is divided by a number a little less than 1, so the answer should be a little more than 120.

Electric Current

Section

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Step 5: Sense


Units are CRITICAL in physics.

It is the units that give meaning to the numbers. 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.

SI Units

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The Système International d’Unités, or SI, uses seven base quantities, also called the fundamental units.

SI Units

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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.

ex: speed is measured in meters per second (m/s)

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

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Measuring standards for kilogram and meter are shown below.

SI Units

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You probably learned in math class that it is much easier to convert meters to kilometers than feet to miles.

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

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

SI Units

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

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A number in the form a x 10n is written in scientific notation where 1 ≤ a < 10, and n is an integer. (An integer is a whole number, not a fraction, that can be positive, negative, or zero.)

Scientific Notation

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When moving the decimal point to the right, you reduce the exponent when using scientific notation.

Right – REDUCE

When moving the decimal point to the left, you make the exponent larger when using scientific notation.

LEFT – LARGER

Common powers of ten include 100 = 1, 101 = 10, 102 = 100, etc.


Write 7,530,000 in scientific notation.

The value for a is 7.53 (The decimal point is to the right of the first non-zero digit.)

So the form will be 7.53 x 10n.

7,530,000. = 7.53 x 106 (Move the decimal point 6 places to the left; exponent gets larger.)

Scientific Notation: Example 1

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1.1


Write 0.000000285 in scientific notation.

The value for a is 2.85 (The decimal point is to the right of the first non-zero digit.)

So the form will be 2.85 x 10n.

0.000000285 = 2.85 x 10-7 (Move the decimal point to the right 7 places; exponent gets

smaller.)

Scientific Notation: Example 2

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1.1


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

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1.1

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 or m/s2, 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.



Section

1.1

example: Calculate the distance a car travels when it is moving at a velocity of 20 meters per second for 10 seconds.

Use the formula: distance = velocity x time ?

Use dimensional analysis: meters = meters x seconds second

Treat the units as if they were algebraic quantities. ?

meters = meters x seconds second

Seconds in the numerator cancel seconds in the denominator.

The formula is therefore dimensionally correct. meters = meters


Dimensional analysis also is used in choosing conversion factors. This is also known as the factor-label method.

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


Section

1.1

1 kg

1.34 kg


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:

Factor-Label Method

Section

1.1

= 1,340 g1000 g

365 days1 year

1 year


Convert 1 year to hours.

Factor-Label Method: Example 2

Section

1.18760 hours/year

1 day

24 hours= 8,750 hours


Pressure is defined as force divided by area of contact ( P = F / A).

What pressure must you apply to your pen in order to create a force of 0.25 N on a piece of paper, if the tip of the pen has a surface area of 3 mm2 touching the paper?

Warmup Problem

Section

1.18760 hours/year


Rules for Rounding Off

1. In a series of calculations, carry the extra digits through to the final answer, then round off. Rounding only occurs ONCE in a calculation!

2. If the digit to be removed is: <5, the preceding stays the same.

example: 1.33 rounds to 1.3 5 or greater, the preceding digit increases by 1.

example: 1.36 rounds to 1.4.

Example: Round 62.5347 to four significant figures.

Look at the fifth figure. It is a 4, a number less than 5. Therefore, simply drop every figure after the fourth, and the number becomes

62.53

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1.1


Definition: All the valid digits in a measurement, the number of which indicates the measurement’s precision (degree of exactness).

Do not count sig figs for non-measurement quantities such as counting (4 washers) or exact conversion factors (24 hrs. in 1 day).

Use the Atlantic & Pacific Rule to determine the sig figs.

Significant Digits (Significant Figures or Sig Figs)

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1.1

PACIFIC

OCEAN

ATLANTIC

OCEAN

The Atlantic /Pacific Rule for Sig Figs

If the…• Decimal is Present

–Count all digits from the Pacific side from the first non-zero digit.

• Decimal is Absent

–Count from the Atlantic side from the first non-zero digit.

–Trailing zeros are indeterminate; they may or may not be significant. Use scientific notation to remove the ambiguity.

Section

1.1 Mathematics and Physics


How many sig figs are there?

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PACIFIC OCEAN

Decimal point present

ATLANTIC OCEAN

Decimal point absent

Count all the digits from the first non-zero digit.

705.00 g 2130 m 523.0 g 706, 000 g0.0098070 mm 9,010, 000 km

5 sig figs4 sig figs5 sig figs

3 sig figs (and 1 indeterminate) 3 sig figs (and 3 indeterminate)

3 sig figs (and 4 indeterminate)


Using the Atlantic rule, we can’t be sure if trailing zeros are significant or not. To specify the exact number of sig figs, use scientific notation.

Exponents do not count towards significant digits.

Example: 9,010, 000 km (3 sig figs and 4 indeterminate).

To write this number indicating:

3 significant digits:



Significant Digits

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1.1

9.01 x 106 km

9.010 x 106 km

9.0100 x 106 km


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

Add / Subtract: Round to the least number of DECIMAL places as determined by the original calculation. (All numbers must be the same power of 10).

Example: 23.1 4.77

125.39 + 3.581

156.841 Round to 156.8 (one decimal place)

Operations with Significant Digits

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Example: Add 5.861 dL + 2.614 L + 3.5 mL

Convert to the same units and power of 10. Add in column form.

5.861 dL = 5.861 x 10-1 L = 0.5861 L

2.614 L

3.5 mL = 3.5 x 10-3 L = 0.0035 L

3.2036 L

Round to the least amount of decimal places 3.204 L

Significant Digits – Addition / Subtraction

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fewest decimal places


Multiplication / Division: Round to the fewest number of SIGNIFICANT FIGURES.

Example:

(3.64928 x 105) (7.65314 x 107)(5.2 x 10-3) (5.7254 x 105) least precise measurement

= (3.64928 x 105) x (7.65314 x 107) ÷ (5.2 x 10-3) ÷ (5.7254 x 105)

= 9.3808 x 109

= 9.4 x 109 because the least precise measurement has 2 sig figs.

Significant Digits – Multiplication / Division

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When doing a calculation that requires a combination of addition/subtraction and multiplication/division, use the multiplication rule.

Example:

slope = 70.0 m – 10.0 m = 3.3 m/s 29 sec – 11 sec

29 sec and 11 sec only have two significant digits each, so the answer should only have two significant digits

Significant Digits – Combination Operations

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Multistep Calculations

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Do not round to 580 N2 and 1300 N2

Do not round to 1800 N2

Final answer, so it should be rounded to two sig figs.

Section Check

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?

Question 1

Section

1.1

A. 1 kg·m/s

B. 1 kg·m/s2

C. 1 kg·m2/s

D. 1 kg·m2/s2

Section Check

Answer: D

Answer 1

Reason:

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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)

Question 2

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A. 2.5×101 m/s

B. 1.5×103 m/s

C. 2.5 m/s

D. 1.5×102 m/s

Section Check

Answer: A

Answer 2

Reason:

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Section Check

Which of the following representations is correct when you solve 0.03 kg + 3.333 kg?

Question 3

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A. 3 kg

B. 3.4 kg

C. 3.36 kg

D. 3.363 kg

Section Check

Answer: C

Answer 3

Section

1.1

Reason: When you add or subtract, round to the least number of decimal places.

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A Physics Toolkit