Measurement and Uncertainty

Question 1

Order of magnitude

Answer 1

Power of 10/Expressing a quality as a plain power of 10 (scientific notation).

Question 2

Mass (kg) ranges of magnitudes of quantities

Answer 2

• Mass of an electron (10^(-32))
• Mass of universe (10^52)

Question 3

Length/Distance (m) ranges of magnitudes of quantities

Answer 3

• Proton (10^(-15)) - diameter
• Universe (10^26) - radius
• Hydrogen atom (10^(-10)) - diameter

Question 4

Time (s) ranges of magnitudes of quantities

Answer 4

• Passage of light across a nucleus (10^(-24))
• Age of the universe (10^20)

Question 5

Density

Answer 5

mass/volume

Question 6

1’s

Answer 6

Distance between one’s hands (m), duration of heartbeat (s), very low temperature (K), 12g of carbon (mol), weight of an apple (N), work done lifting an apple (J)

Question 7

Fundamental units

Answer 7

The basic units from which all others are derived: M(meter - distance, m)y K(kilogram - mass, kg)id S(second - time, s)ister K(kelvin - temperature, K)icked A(ampere - electric current, A) Mole (mole - amt of substance, mol).

Question 8

Derived units

Answer 8

Combos of fundamental units: ex. Newton (force, N [F = ma → kgms^(-2)]), Joule (energy, J [W = Fd → kgm^(2)s^(-2)]), Watt (power, W [P = W/t → kgm^(2)s^(-3)]).

Question 9

Converting (same unit)

Answer 9

Where you see the metric multiplier, just plug it in: if it’s ex. squared or cubed, you have to square/cube the multiplier. If a different unit entirely, just multiply (proceed/ignore).

Question 10

Error

Answer 10

• Occurs when there’s a difference between an obtained value and the accepted value (will always be a present [question is how much]: there has to be some comparison for you to say there’s an error)
• Two types: random and systematic

Question 11

Random error

Answer 11

• Caused by any factor that affects the measurement of a variable; by chance
• Always present in measurements and cannot be completely eliminated, but can be reduced by repeated readings/measurements
• Identified on a graph through scattered points
• Sources: readability of instruments (or calibrations [ex. meter stick calibrations that are old and difficult to read]), weather change, etc.

Question 12

Systematic error

Answer 12

• There’s consistency in the shifted data
• Also called zero error b/c instrument looks (is?) shifted away from zero (wear, shipment, etc.)
• Caused by any factor that consistently affects the measurement of a variable
• Cannot be reduced by repeated readings
• Identified on a graph as best-fit line not starting from the origin (shift can be on the x- or y-axis) and points organized on the line
• Sources: instruments w/ zero error, wrongly calibrated instruments (ex. meter stick w/ misprint)
• Usually found in instruments such as watches, thermometers, ammeter, etc. (anything w/ a pointer)

Question 13

Sig fig counts for calculated values

Answer 13

• When multiplying/dividing/raising to a power/taking a root, as many sig figs as least precisely known number entering calculation
• When adding/subtracting, equal to the least # of decimal places in the numbers added/subtracted
• Use rules for rounding for either

*See notebook for exception

Question 14

Stoichiometry

Answer 14

Multiply by 1 (denom should be same unit, but equal to num, which should be the unit would be what we’re converting). Remember: to get rid of ex. hrs in denom and s in num, multiply by 1h/60 min and 1 min/60s.

*See review

Question 15

NOTE

Answer 15

• When lone 10’s, think of them as 1 x ___: they should have the same number of zeros (including the one before the decimal for negative powers) as their exponent
• Coefficient should be between 1 (inclusive) and 10
• Diameter of a hydrogen nucleus is the same as the diameter of a proton
• If problem says estimate, you can use 3 for pi
• As long as it’s a nucleus, assume 10^(-15): even if there are more, order of magnitude would still be 10^(-15), no matter the coefficient
• No slashes (use negative powers)
• Use units/formulae
• vector x scalar = vector (but cannot be added b/c different dimensions)
• ex. metal metre ruler expans in hot weather (too-small measurements), don’t consider friction

*See past questions, notes for uncertainty, sf

Question 16

Uncertainties

Answer 16

• Uncertainties are errors
• Value after ± represents error present (absolute error/uncertainty)
• Measurements are usually reported w/ uncertainties
• Can be reported in three ways: absolute uncertainty, f- fractional uncertainty, and percentage uncertainty
• The actual reading is the value that comes before the ±

Question 17

Calculating fractional and percentage uncertainty

Answer 17

• Fractional is absolute uncertainty divided by the actual reading (decimal)
• Percentage is fractional written as a percent value (multiplied by 100)

Question 18

Combining uncertainties in results

Answer 18

• For addition/subtraction, add the absolute uncertainties
• For multiplication/division, add the fractional/percentage uncertainties and then report the final answer w/ the absolute uncertainty using:
  Total Fractional = Absolute/Actual
• When raising to a power, multiply by power raised to (b/c we’d be adding the same value that many times)
• When reporting actual w/ absolute, follow rule of decimal places (not sig figs)

Question 19

Error bars

Answer 19

• Uncertainties shown on a graph using short vertical/horizontal lines w/ bars
• Ends of bars represent maximum and minimum values of plotted data
• Same distance
• If uncertainties are negligible, may not show (usually, one axis [variable] will have minute uncertainty)

Question 20

Finding the slope of a straight line (on a graph)

*What does the slope represent if the x-axis represents time?

Answer 20

• For the slope/gradient/rate of change of a straight line, a triangle is used (must be more than half of the line) then change in y over change in x
• If directly proportional (goes through origin and is a straight line), you don’t have to draw a triangle (just y over x)
• Unit of slope is unit of y over unit of x
• If the x-axis represents time, slope represents rate at which the y-axis changes

Question 21

Finding the slope of a curve (on a graph)

Answer 21

• For a curve, if you’re asked to find the slope, a tangent must be drawn (just make it meet both axes [must again take up at least half the graph], then draw an imaginary line from your x-point to the tangent/point—that’s your triangle, so just use change in y over change in x)

Question 22

Variable relationships (on a graph)

Answer 22

• Proportional if line straight and passes through origin
• Linear if line straight, but there’s a y-intercept
• Inversely proportional if y decreases at a constant rate (straight and going downwards)
• Inverse if y decreases at a changing rate (curved and going downwards)

Question 23

Using the straight line equation

Answer 23

• Straight line equation written in the following forms: y = mx + b, y = b + mx
• No b means no y-intercept, so proportional
• When you draw triangle, slope would go inside
• Could become inverse if x in denominator (ex. directly proportional to r^(-2) and inverse [?] w/ r^2)

Question 24

Best-fit lines

*Exponential curves?

Answer 24

• Best-fit lines not necessarily straight
• Points must be evenly distributed
• If error bars are involved, best-fit line must pass through all error bars (again, can be a curve [and can be maybe forgiven if just one out of place)
  *Exponential curves never touch the x-axis

Question 25

Maximum and minimum gradient lines

Answer 25

• For max gradient lines, must touch smallest of first and biggest of last
• For min, must touch biggest of first and smallest of last
• On y-axis, can see uncertainties of y-intercept (max and min respectively [OPPOSITE]— (max + min)/2 for abs uncert of y-intercept)

Question 26

Calculating the absolute uncertainty for the best-fit line (error bars involved)

Answer 26

1. Draw your max and min gradient lines
2. Find the slope of each of the lines (assumed that you have the slope of your best-fit line)
3. Use this equation:
   best-fit slope uncert = (max slope - min slope)/2
4. Report final ans as slope ± uncert

Question 27

NOTE

Answer 27

*Always finish your processing (don’t use fractions)
*Delta symbol represents uncertainty
*See packet for formulas
*For ex. 283.64, uncert. is ± 0.01; for ex. 283.6, uncert. is ± 0.1 (more sig figs = more precise)
*Calculate actual, calculate individual uncertainties, add to find total fractional, find absolute
*Constants irrelevant; ignore them
*When given in %, that’s your fractional (percentage?)
*Accounting for negative values
*If something was 0, wouldn’t include (?), so…
*Order exactly (so if unsure about which is which, look at form)
*For circular motion formula, substitute F w/ type of force
*See notes for log
*Just think of as negative, honestly
*No pen for drawing graphs

Question 28

Scalar and vector quantities

Answer 28

Scalar quantity: a quantity that has only magnitude (no direction [ex. mass, speed, all forms of energy, distance, time, potentialrs, ampere, pressure, etc.])

Vector quantity: A quantity that has both magnitude and direction (ex. velocity, displacement, acceleration, all fields, weight/force, etc.)

Question 29

Multiplication of a vector by a scalar

Answer 29

If vector a (arrow/dash on top means vector, but can’t type) is multiplied by a # k, then:
Ka (arr. on a) has same direction as vector a if k is greater than 0
Ka (arr) has opposite direction to vector a if k is less than 0
*Division is the inverse (y/2 is the same thing as y x ½): just treat divisor as a scalar between 0 and 1

Question 30

Adding vectors

Answer 30

Two methods: parallelogram (bring tails together, complete parallelogram, and draw resultant arrow from intersection to other point of parallelogram [tail at intersection]) and head-to-tail (head to tail and connect with tail of resultant with tail of other)
For subtraction, add opposite (flip direction)
Independent of order
If more than one, continue process

Question 31

Finding components

Answer 31

We may need to take a vector and find what other vectors were added together to produce it: in most cases, involves determining the perpendicular components of a single vector (ex. x and y, north-south, east-west components)
When given a slanted vector and asked to find the horizontal or vertical components, (see notebook: F sin theta is vertical component and F cos theta is horizontal component [theta and theta go in corder, connect upper and side to form box, right angles in bottom right and top left corners, force going outward [dashed and bold lines same, so sine])

Question 32

NOTE

Answer 32

Use Pythagorean theorem for magnitude and trig relations for angle
If arrows form a line and are going in the same direction, will get sum (you’’ get a 0 if you do something else)