www.lsmsa.edu teacher Robert Dalling's Physics Lectures (see also www.ushumans.net)
Measurement
Science consists of facts and understandings obtained from repeatable measurements (experiments). Measurements always have a unit–for example meters–and a numerical value consisting of a string of measured digits. A measured digit is termed a significant figure. There is always an experimental uncertainty in the last measured digit. This uncertainty is propagated through all subsequent calculations. Results are often summarized graphically.
A significant figure is a measured digit. For example, you might measure the length of a pencil to be 15.25 cm. This measurement has 4 significant figures. Significant figures are tracked because we want to know if theory and measurements agree to all significant figures. A measured value is deemed accurate if it agrees with the accepted value. The measurement having the largest number of significant figures is the most precise. p24. Parallax occurs when you read a meter from the side rather than head on.
Nonzero figures are significant, as are all zeroes after the decimal point or between other significant figures. Zeroes used as placeholders are not significant.
number significant figures
1030 3
260,000 2
260,300 4
2.010 4
0.0023 2
0.000058 2
Arithmetic of significant figures
When adding or subtracting numbers, the result should not be carried beyond the first uncertain decimal place. For example, 8.16 + 74 = 82 not 82.16.
When multiplying or dividing numbers, write the result with the same number of significant figures as occurred in that having the fewest significant figures. For example, 8.3 x 1045 = 8700 not 8672.5.
CQ:
22+430+79=530
310-17=290
15.1+1.24+107=123
14x219/330 = 9.3
The number one million is written 1,000,000. To save us from writing such strings of zeroes, this number is abbreviated 1.0 x 109. The exponent ‘9' here means that the number 1.0 is followed by nine zeroes. The leading number contains all significant figures available. Numbers less than one are written with negative exponents. Let’s write the number 1 surrounded by zeroes: 000001.000. Point at the ‘1' and say out-loud “zero.” Then count leftward 1, 2, 3 ... or rightward -1, -2, -3 ... to identify the exponent or powers of ten. We start with 100 and move leftward to get 101, 102, 103, ... or we move rightward to get 10-1, 10-2, 10-3.
NASA calculates the number of stars in the universe and the number of molecules in a cubic inch of water.
Here is a review of the rules of exponents and logarithms.
Student:
Zoom through the powers of ten, from the outer reaches of the universe down to elementary particles.
The basic units involve measurements of time in seconds, mass in kilograms, length in meters, electrical current in amperes, temperature in Kelvin, amount of matter in moles, and luminous intensity in candela. Every other unit of measurement is a combination of these five. The world is full of traditional measures such as inches, feet, and cubits.
We can add together only things having the same dimension. For example, we can add 3 meters plus 12 kilometers but we can’t add 3 meters plus 4 kilograms.
We use predefined conversion factors to convert an amount from one unit to another. While doing so, we multiply and divide units just as any other algebraic quantities. For example, the out of date unit “inch” is defined through the conversion factor 1 inch = 0.0254 meter = 2.54 cm. How many meters are in 100 inches?
( 100 inches )( 0.0254 meter / inch ) = 2.54 meters
Notice that units cancel as algebraic symbols.
How many kilometers are in one mile?
(1 km/ 1000 m)(1 m / 100 cm)(2.54 cm/ inch)(12 inch/ft)(5280 ft/mile) = 1.609 km/mile
CQ: How many inches are in 35 cm?
(35 cm)( 1 inch/2.54 cm) = 13.8 inches.
Conversion factors are squared or cubed when converting areas or volumes. For example, the number of square centimeters in 6.2 square inches is found from
( 6.2 in2 )( 2.54 cm / inch )2 = 40 cm2. See that the value 2.54 was squared.
CQ: How many cubic centimeters are in 100 cubic inches?
( 100 in3 )( 2.54 cm/inch )3 = 1640 cm3.
Example:
The units of the metric system are defined in terms of the properties of water. The mass of one gram of water is that contained in a volume of one cubic centimeter. In terms of kilograms and meters, mass of water per unit volume is
( 1 gram / cm3 )( 1 kg / 1000 g )( 100cm/m )3 = 1000kg/m3.
JC p24 # 1.52.
A coal-fired, electrical power generating plant burns 100-train-cars of coal per day. If 10% of this volume is ash, what is the volume of ash produced in one year? The lengths of the sides of a train car result in a volume of 42x8x8 = 2688 ft3.
Ash-Volume/year = ( 100 cars/day )( 2688 ft3 / car)( 365 days / year )( 0.1 ash ) = 9.8x106 ft3 of ash per year.
In cubic meters, this is volume is
( 9.8x106 ft3 ) ( 12 inch / ft )3 ( 2.54 cm / inch )3 ( 1 m / 100 cm )3 = 280,000 m3. Which is a cube having sides of length 65 meters. Some plants burn twice this amount of coal, while a nuclear-powered, electrical generating plant burns none.
T p16 # 23. Given one cell / 7x10-9 m. Calculate cells / inch.
( 1 cell / 7x10-9 m )( 1 m / 100 cm )( 2.54 cm / inch ) = 4 x 106 cell/inch.
T p17 #40. Air pressure of 14.7 lbs/in2 acts on the entire surface of the Earth of radius 3960 miles (6370 km). What is the total weight of the air? (The area of a sphere is 4 π r2.)
( 14.7 lbs/in2 ) ( 12 in/ft )2 ( 5280 ft/mile)2 4π(3960 mile)2 = 1.16 x 1019 lbs.
With each breath you take, how many of the “air molecules” you take in were exhaled by Caesar’s last breath? About 1.
Example:
There are about 5 x 1010 galaxies in the universe, 5 x1021 stars in each galaxy, but 6 x1023 water molecules in one cubic inch of water.
Graphing rules: use the entire page, start each axis at zero, label each axis and include units, include error bars, write the expression for the curve. Draw the best fit line through the data points. When calculating the slope use widely spaced points taken from the line–not from the data points–and indicate which points were used in determining the slope. Draw minimum and maximum slope lines to calculate the range in the slope.
For a straight line, y = mx + b,
where b is the y intercept and
m = slope = rise / run
Graph
time (seconds) =0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0
distance (± 0.5)=1.2, 2.0, 2.8, 3.7, 5.1, 5.8, 7.0, 8.1, 8.9, 10.4 cm
Choose axes, scales, plot points with error bars, draw a best-fit line, calculate the slope and its range, find the intercept, write the equation of the line.
Experimental errors
Tycho Brahe 2 (1546-1601), whose dueled-away nose was replaced with gold and silver, measured the positions of planets by carefully pointing wooden sticks at them each night and then measuring the angle those sticks made with the ground. His measurements were accurate to 3%. Kepler 2 (1571-1630) used this data to determine that the motion of Mars around the Sun differed from being circular by 7%. He knew Tycho's measurements were accurate to 3% so the 7% departure could not be explained away. The orbit of Mars is an ellipse that, in this case, is a circle bent by 7%.
Scientific results often include estimates of the uncertainty in the conducted experiments. This is done because there is always an uncertainty in the last measured digit whenever one makes a measurement. For example, measure the length of a pencil with a ruler. The ruler's smallest division might be one-sixteenth of an inch (or it might be one-tenth of a centimeter) and the end of the pencil might fall about one-third of the way along one of those smallest divisions. But it is hard to tell exactly because sometimes it looks like it may be one-quarter of a division instead of one-third of a division. The usual rule-of-thumb is to take the experimental uncertainty to be one-half of the smallest division (interpolating between the lines )of your analog meter or measuring instrument or the smallest division of a digital meter. The measurement of the length of the pencil might be written as 6.33 plus or minus (±) 0.05 centimeters–that is, its length is measured to be within 0.05 cm of 6.33 cm. In the same way, you can read your car's speed meter, and your bathroom scale, accurately to about half the smallest division of the scale. If you weigh yourself before, and again after, drinking a glass of water then you will find that the scale has trouble distinguishing that change in weight. Most scientific measurements are accurate to about 0.1%, but some are accurate to one part in one hundred billion. (In comparison, one hundred billion seconds is a 30,000-year time span. It's hard to imagine measuring a 30,000-year time span and being accurate to within one second.)
Scientists are concerned the accuracy of their measurements for a few reasons. They want to compare their results with those of other scientists, and they want to compare their measurements to the numbers obtained from theoretical equations. For example, if one can measure accurately to 0.1% then the results are expected to be within 0.1% of theoretical predictions or the accepted value obtained from prior measurements; otherwise, something is wrong with either the theory or the experiment. Another use is that if one can measure accurately to just 1% then it is meaningless to worry about affects accounting for less than 1% of the system’s development. This is the reason we can ignore the effects of air resistence in the motion of a rock dropped to the ground.
Most hand-made items can be built within an accuracy of about 5%. This is often close enough. The number pi has the value 3 when written with that accuracy.
When we measure something we take an experimental error or uncertainty in the last measured digit to be one-half the smallest division of the meter. Sometimes we record a larger experimental error because of circumstances–poorly defined edges, for example.
WE ALWAYS WRITE THE MEASUREMENT AND ITS ERROR TO A CONSISTENT DECIMAL DIGIT:
write 1.34 ± 0.05
not 1.345 ± 0.05
nor 1.3 ± 0.05.
OTHERWISE, THE WORLD WOULD END.
Measure the following items, write down the experimental error and write a sentence to explain why you chose that uncertainty.
1) width of your
2) height, length, and width of a table
3) width of your wrist
4) diameter of a ball
Propagation of experimental errors
If a single measurement is repeated hundreds of times, then its standard deviation can be calculated. The calculus of differentials can be used to relate the error in a calculation that combines measurements. We'll use the "weakest link" rule in that when adding or subtracting measured numbers we propagate the largest error--when multiplying or dividing we propagate the largest fractional error.
For L ± Δl = 7.6 ± 0.2 meter and W ± Δw = 12.2 ± 0.1 meter we get L + W = (7.6 + 12.2) ± 0.2 = 19.8 ± 0.2 meter.
For L ± Δl=7.6 ± 0.2 meter and W ± Δw = 12.2 ± 0.1 meter we get Δl/L = 0.2/7.6=0.026 and Δw/W=0.1/12.2=0.0082. The fractional error in L is larger than that of W. Then temporarily A = LW = 7.6x12.2 = 92.72 square meters. We want ΔA/A = 0.026 so ΔA = 0.026 x A = 2.54072. We always round propagated errors to one significant digit and then round the calculation to a consistent digit. The final answer is A ± ΔA = 93 ± 3 square meters.
Calculate table circumference = length plus width
Multiply table width times length to obtain its area.
Speed = distance / elapsed time.
Measure the speed of a person walking across the room. Write down the experimental errors in the distance and time. Propagate the largest fractional error by making the fractional error in the result equal to the largest fractional error in the distance or time.
Notice that 1.01²=1.02, which means that a 1% difference becomes 2%. In general, a measurement m and its error e raised to the nth power gives (m ± e)n = mn ± ne
Propagate errors in the following calculations:
We conduct an experiment (that’s the “purpose” you’ll state in the lab report) in physics because we want to 1) test a theory/equation, e.g. F=ma, 2) measure a physical quantity, e.g. the speed of sound, or 3) experimentally deduce a relation between two quantities. We vary one quantity and measure one other (that’s the lab “procedure”). We either state the measured value or relationship or we decide if the theory is correct by whether or not it predicts the measurements (that’s the lab “conclusion”). Every equation and physical quantity in the textbook was found this way.
The lab report contains:
1) Title, date, partner
2) Purpose. Describe the overall purpose of the experiment. For example, “Measure g.”
3) Sketch the apparatus with the parts labeled. Include model numbers. Indicate in the sketch the meaning of the symbols used. For example, show string length ‘L’ along the sketched string.
4) Procedure. Write down what was varied, what was measured, and how it was measured. Describe any observations made or difficulties encountered. You want to be able to pick up your lab notebook a year from now and be able to repeat your measurements and understand what steps were done. In you own words, say “while varying ___ we measured ___ with a ___, having an experimental error of ___ due to __.” For example, “while varying its string length, we measured the period of a pendulum with a timer having an experimental error of 0.5 s due to human reaction-time. Repeat that sentence for each measured quantity.
5) Data table (for organization). Record measurements directly into the book–not onto scratch paper.
6) Record the experimental error in every measurement, typically interpolating to one-half the finest division of the meter, and propagate those errors through subsequent calculations. Write down the reason for choosing the size of each error. For example, you might state “The error in the measured string length was taken to be ± 0.5 mm due to interpolation,” or “± 2mm due to its indistinct edge.” The measurement and its error should be written to a consistent decimal digit: 12.345 ± 0.005 not 12.34 ± 0.005 nor 12.345 ± 0.0005. Propagate the largest error when adding measurements, and the largest fractional error, Δm/m, when multiplying. Use (m ± Δm)n = mn ± nΔm and r(m ± Δm ) = rm ± rΔm. For cos( m ± Δm), use the one-significant-figure range in cos(m+Δm) and cos(m-Δm). If we can measure accurately to 1% then we expect to be within 1% of accepted values and we can safely ignore the effects of any phenomenon accounting for less than 1% of the system’s development because we couldn’t detect its effects anyway.
7) Calculations. Show just a sample calculation when it is repeated many times.
8) Graphs. Use the entire page, start each axis at zero, label each axis and include units, include error bars, write the expression for the curve. Draw the best fit line through the data points. When calculating the slope use widely spaced points–not data points–taken from the line and indicate which points were used in determining the slope. Draw minimum and maximum slope lines to calculate the range in the slope. Statistically, one-third of the data points should lie farther than one error bar from the average line. State the physical meaning of the slope and intercept.
9) Results of the measurements and calculations. Compare measurements to theory.
10) Explain discrepancies between theory and measurement. Write down a few ways in which the inefficient apparatus or idealizations in the theory might cause a discrepancy. Indicate whether the effect would raise or lower your result. If we measure g=9.5 ± 0.2 m/s2 then the experimental error is 0.2 and the discrepancy is 0.3.
11) Conclusion about the theory you tested, the quantity measured, or the relationship found.
12) Answer the lab questions.
Be unambiguous and concise. One sentence for each of the purpose, procedure, and conclusion. Always ask yourself if your measurements and results make sense.
Things you get out of the lab course:
1) Its immediate value is in making the physics more apparent when it happens in real life right in front of you rather than just reading about it in the textbook. The meaning of the symbols become more apparent when you measure them. Don’t rush through the lab taking data as fast as possible just to get to the final result. Stop to think about the physics that is occurring.
2) The lab notebook. Being able to record measurements in an organized manner directly into a lab book is a basic requirement for doing experimental research. We are learning what graphs and experimental errors tell us.
3) The format of the lab writeup is standard.
WE ALWAYS WRITE THE MEASUREMENT AND ITS ERROR TO A CONSISTENT DECIMAL DIGIT–for example,
write 1.34 ± 0.05
not 1.345 ± 0.05
nor 1.3 ± 0.05
OTHERWISE THE WORLD WOULD END.
Problem Solving Procedure in Physics
Every beginning student of physics tries at first to solve problems by reading the question, grabbing a pencil, and writing 3 cos( 24 ) + 7 sin( 15 ). But this approach rarely works as it requires you to solve the entire problem in your head. This causes unnecessary confusion and frustration. Through future years of study, problems get increasingly long: Each of this year’s problems take 10 minutes to solve, next year’s take one hour, and the problems of the years after that take 10 to 100 hours to solve. We can’t solely plan within our heads the blueprints for a building. We all learn the hard way that we are more successful at solving problems if we follow an organized approach.
1) Draw a picture to explain the problem and indicate the axes and their origins in the sketch (unless the problem involves nothing more than plugging numbers into an equation). This helps keep arithmetic signs consistent throughout the solution. Indicate the meaning of the symbols in the sketch. For example, when h = building height, show ‘h’ next to the drawn rectangle.
2) List given and unknown quantities, and including units. For example, F = -3 N, m = 3 kg, a = ?
3) Describe your physical reasoning in a written sentence. When you review your solution in the future, you'll be grateful that you had. For example, “the force of ground friction slows the box.”
4) Write equations relating the known and unknown quantities. For example, F = ma.
5) Algebraically solve equations for unknowns before plugging in any numerical values. For example, a = F / m. Show algebraic steps.
6) Plug the numbers into your final algebraic equation and then box your final answer. For example, a = F / m = -3 N / 3 kg = -1 m/s2.
7) Make sure your answer is reasonable and that it has the correct dimensions.
Test preparation accumulates each day
Make notes to yourself that will teach yourself, help you prepare for tests, and help when you review the material during future courses. In the first few pages of your notebook, accumulate a list of physical quantities and equations.
1) Write down the definition of each physical quantity encountered in the course, including velocity, force, and electric charge and such. Note each quantity’s standard algebraic symbol, its SI units, and whether it is a scalar or a vector quantity. Indicate typical and extreme numerical values and explain the meaning of positive, zero, and negative values. Note any alternative units.
2) Write down each equation relating physical quantities. Explain the equation in words, define each symbol, and describe when this relation is useful. Show a typical example of its use. Note any restrictions on its use–for example, “when air friction accounts for less than 1% of the system’s time evolution,” or “under isobaric development.”
There are many online collections of problems and their solutions, for example, http://zebu.uoregon.edu/~probs/probm.html
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