Introduction to Physics Chapter 1

Introduction Physics is the most basic of the sciences Deals with the behavior and structure of matter Topics in Physics: Motion, fluids, heat sound light, electricity and magnetism, relativity, atomic structure, condensed-matter physics, nuclear physics, elementary particles, and astrophysics.

1-1 Science and Creativity In general, the goal of all of the fields of science is to attempt to understand the world around us by exploring it in an orderly way. Science is often times a creative activity Example: observations and inferences Aristotle (384-322 B.C.) observed that objects given a push along the ground will slow to a stop → he inferred that the natural state of objects is to be at rest. Galileo (1564-1642) imagined that if friction was eliminated, an object put in motion would move indefinitely

1-1 Science and Creativity Theories are created to explain observations Scientific theories can be viewed achievements, but they do require testing through experimentation to gain support Theories are never absolutely verified – they are not “proved” Theories can be modified or replaced as more is learned about natural phenomena Copernicus’s Sun-centered theory of the universe vs. Ptolemy’s Earth-centered theory

1-1 Science and Creativity Theories are most valuable when they are simpler (or no more complex) than previous theories and unify and explain a greater variety of phenomena These principles, in addition to quantitative agreement, influence the acceptance of a theory Einstein’s theory of relativity differs very little from previous theories from Galileo and Newton in many situations – however, it has drastically altered our view of space-time and the relationship between mass and energy (E=mc ) 2

1-2 Physics and Other Fields It is not until the last couple of centuries that distinctions were made between different fields of science The field of science that we consider as physics has been influenced by other fields (not just scientific fields) and continue to influence other fields.

1-3 Models, Theories, and Laws Model – a kind of analogy or mental image of the phenomena in terms of something we are familiar with Models provide a mental or visual image when you cannot actually see what is happening Examples: wave model of light; atomic models Models vs. Theories

1-3 Models, Theories, and Laws Law – a concise but general statement about how nature behaves Example: Law of Conservation Energy Laws can sometimes take the form of a relationship or equation between quantities Example: Newton’s equation F=ma

1-3 Models, Theories, and Laws Like theories, we can never be sure that laws are absolutely true – when new information is learned, laws can be modified or discarded Theories – explain Laws – describe (without explanation)

1-4 Measurement and Uncertainty In order to understand the natural world, scientists seek to find relationships between quantities Examples: How does the magnitude of a force affect an object’s acceleration? How does a change in temperature affect the pressure of a gas in a closed container?

1-4 Measurement and Uncertainty Accurate measurements are an important part of physics, but no measurement is absolutely precise There is an uncertainty associated with every measurement Uncertainty arises from various sources Individual errors → Example: misusing equipment The limited accuracy of every measuring instrument

1-4 Measurement and Uncertainty The number of reliably known digits in a number is called the number of significant figures Significant figures in a measurement include all of the digits that are known precisely plus one last digit that is estimated 23.21 → 4 significant figures 0.062 → 2 significant figures

Significant Figures - Rules 1. Every nonzero digit in a recorded measurement is significant. - Example: 24.7 m, 0.743 m, and 714 m all have three sig. figs. 2. Zeros appearing between nonzero digits are significant. - Example: 7003 m, 40.79 m, and 1.503 m all have 4 sig. figs.

Significant Figures - Rules 3. Zeros appearing in front of all nonzero digits are not significant; they act as placeholders and cannot arbitrarily be dropped (you can get rid of them by writing the number in scientific notation). - Example: 0.0071 m has two sig. figs. And can be written as 7.1 x 10-3 4. Zeros at the end of the number and to the right of a decimal point are always significant. - Example: 43.00 m, 1.010 m, and 9.000 all have 4 sig. figs.

Significant Figures - Rules 5. Zeros at the end of a measurement and to the left of the decimal point are not significant unless they are measured values (then they are significant). Numbers can be written in scientific notation to remove ambiguity. - Example: 7000 m has 1 sig. fig.; if those zeros were measured it could be written as 7.000 x 103

Significant Figures - Rules 6. Measurements have an unlimited number of significant figures when they are counted or if they are exactly defined quantities. - Example: 23 people or 60 minutes = 1 hour * You must recognize exact values to round answers correctly in calculations involving measurements.

Significant Figures in Calculations In calculations involving measurements, an answer cannot be more precise than the least precise measurement from which it was calculated. Example: The area of a room that measures 7.7 m (2 sig. figs.) by 5.4 m (2 sig. figs.) is calculated to be 41.58 m2 (4 sig. figs.) – you must round the answer to 42 m2

Significant Figures in Calculations The answer to an addition or subtraction problem should be rounded to have the same number of decimal places as the measurement with the least number of decimal places. Example: 34.61m – 17.3m = 17.31 → 17.3 (1 decimal place)

Significant Figures in Calculations In calculations involving multiplication and division, the answer is rounded off to the number of significant figures in the least precise term (least number of sig. figs.) in the calculations Example: 8.3m x 2.22m = 18.462 → 18m Example: 8432m ÷ 12.5 = 674.56 → 675m 2

Scientific Notation scientific notation – a number is written as the product of two numbers: a coefficient and a power of ten Example: 36,000 is written in scientific notation as 3.6 x 104 or 3.6e4 Coefficient = 3.6 → a number greater than or equal to 1 and less than 10. Power of ten / exponent = 4 3.6 x 104 = 3.6 x 10 x 10 x 10 x 10 = 36,000

Scientific Notation When writing numbers greater than ten in scientific notation the exponent is positive and equal to the number of places that the decimal has been moved to the left.

Scientific Notation Numbers less than one have a negative exponent when written in scientific notation. Example: 0.0081 is written in scientific notation as 8.1 x 10-3 8.1 x 10-3 = 8.1/(10 x 10 x 10) = 0.0081 When writing a number less than one in scientific notation, the value of the exponent equals the number of places you move the decimal to the right.

Accuracy and Precision Good measurements in the lab are both correct (accurate) and reproducible (precise) accuracy – how close a single measurement comes to the actual dimension or true value of whatever is measured precision – how close several measurements are to the same value

Accuracy and Precision All measurements made with instruments are really approximations that depend on the quality of the instruments (accuracy) and the skill of the person doing the measurement (precision) The precision of the instrument depends on the how small the scale is on the device. The finer the scale the more precise the instrument.

SI Units The International System of Units, SI, is a revised version of the metric system Correct units along with numerical values are critical when communicating measurements. The are seven base SI units (Table 1-2) of which other SI units are derived (derived units). Sometimes non-SI units are preferred for convenience or practical reasons

Common SI Prefixes Units larger than the base unit Tera T Giga G terameter (Tm) Giga G e-9 = 0.000000001 gigameter (Gm) Mega M e-6 = 0.000001 megameter (Mm) Kilo k e-3 = 0.001 kilometer (km) Hecto h e-2 = 0.01 hectometer (hm) Deka da e-1 = 0.1 decameter (dam) Base Unit e0 = 1 meter (m)

Common SI Prefixes Units smaller than the base unit e0 = 1 Deci d meter (m) Deci d e1 = 10 decimeter (dm) Centi c e2 = 100 centimeter (cm) Milli m e3 = 1000 millimeter (mm) Micro μ e6 = 1,000,000 micrometer (μm) Nano n e9 = 1,000,000,000 nanometer (nm) Pico p e12 = 1,000,000,000,000 picometer (pm)

Conversion Factors The same quantity can usually be measured or expressed in many different ways. Examples: 1 dollar = 4 quarters = 10 dimes = 20 nickels = 100 pennies 1 m = 10 dm = 100 cm = 1000 mm Whenever two measurements are equal, a ratio of these measurements will equal one Ratios of these equivalent (equal) measurements are called conversion factors.

Conversion Factors In a conversion factor, the measurement in the numerator (on the top) is equivalent to the measurement in the denominator (on the bottom. Example: “one meter per 100 centimeters” Conversion factors are defined or exact quantities, which means they have an unlimited sig figs – your starting value determines your significant figures 1 m 100 cm

Conversion Factors Because conversion factors are equal to one, when a measurement is multiplied by a conversion factor, the size of a measurement stays the same. Example: How many meters are in 0.68 km? 1. Unknown: 0.68 km in meters 2. Known: - 0.68 km - 1 km = 1000 m → Conversion Factor: 3/4. Solution-Calculation: 1000 m 1 km 0.68 km x 1000 m = 680 m 1 1 km

Dimensional Analysis In dimensional analysis you use the units (dimensions) that are part of measurements to help solve (analyze) the problem When using conversion factors in dimensional analysis always use the conversion factor that has the unit of the known in the denominator This cancels the known unit and gives the answer in the unknown units

Displaying Data - Graphing A well-designed graph can give you more information than either words, columns of numbers, or equations However, for this to be true, the graph has to be drawn properly.

Displaying Data - Graphing An important step in analyzing data is to identify the independent variable (or manipulated variable) and the dependent variables (or responding variables). Independent variable – altered by the experimenter; influence other variables Dependent variables – possibly change as a result of changes in the dependent variable.

Displaying Data - Graphing See Table 2-2 an Figure 2-11 (handout) What is the independent variable? What are the dependent variables? What is the relationship between the independent variable and the dependent variables?

Linear Relationships m is the slope b is the y-intercept When there is a linear relationship between two variables a graph of there relationship will be a straight line (see Figure 2-13). This relationship can be written as a linear equation → y = mx + b m is the slope b is the y-intercept Both are constants that can be found on the graph

Linear Relationships Slope (m) – the ratio of vertical change (∆y) to horizontal change (∆x) m = rise/run = ∆y / ∆x To calculate slope choose two points, A and B, as far apart on the line as possible ∆y is equal to the difference in vertical values of A and B; ∆x is equal to the difference in horizontal values of A and B Example (2-13): m = (20-0)m / (27-0)m/s = 20m / 27m/s = 0.74s

Linear Relationships y-intercept (b) – the point at which the line crosses the y-axis; it is the value of y when x is zero Example (2-13): When x is zero, the value of y is zero → in cases like this, the equation becomes y = mx If y gets smaller as x gets larger, then ∆y / ∆x is less than zero and the slope is negative Example:

Quadratic Relationships When a smooth line drawn through the data points curves upward (not a straight line), the graphs are frequently parabolas This indicates that the variables are related by the equation: y = kx This equation is one form of a quadratic relationship k is a constant that shows you how fast y changes with x See Figure 2-14 2 2

Inverse Relationships Sometimes variables have an inverse relationship and form a hyperbola when graphed. The general equation for an inverse relationship is xy = k or… y = k(1/x) = kx -1

Manipulating Equations The way in which quantities/variables relate to each other can be represented symbolically by an equation, as well as a graph. Example: The relationship between the circumference of a circle and the circle’s diameter can be written as C = πd, or it can be represented as a linear graph You can use the rules of algebra to rearrange equations C = πd → d = C/ π

Solving Equations Using Algebra From the graph of reaction distance versus speed → reaction distance = d; speed = v; and slope (which had units of time) = t This relationship can be represented by the equation: d = vt or d = tv d = tv has the same form, but different symbols then y = mx You can solve for distance if you know the speed and time, but what if you know d and t and want to solve for v?

Solving Equations Using Algebra When rearranging equations using algebra, any process that you do to one side of the equation you must do to the other side of the equation Example: d = vt → Solve for v. The steps can be performed in any sequence, but make sure you perform the same operations on both sides of the equation.

Solving Equations Using Algebra Example: Solve the following equation for x. ay/x = cb/s Example: Solve the following equation for x. y = mx + b

Units in Equations Most physical quantities have units and numerical values When using quantities in equations you must substitute the numerical value and the units Mathematical operations can be done on units just as they can be done on numbers Make sure to use the same units for the same type of measurement Example: If a problem involves length measurements in meters (m) and centimeters (cm) convert them both to the same units

Units in Equations Example: Solve for d using d = vt. v = 11.0 m/s and t = 6.00 s