Chapter 1
1.1 Standards of length, mass, & time 1.2 Matter & Model Building 1.3 Density & Atomic Mass 1.4 Dimensional Analysis				1.5 Conversion of Units 1.6 Estimates & Order of Magnitude Calculations 1.7 Significant Figures & Uncertainty Propagation 1.7 (121) Scalars & Vectors
1.1 Standards of length, mass, & time
The meter The meter originally defined as one ten millionth of the distance from a longitudinal line that passes through Paris to the North Pole. Current definition of a meter: the distance traveled by light through a vacuum during 1 / 299,792,458 of a second, which defines light to be approximately 3 x 10^8 m/s.			The kilogram The kilogram (kg) is defined as the mass of a specific Pl-Ir alloy cylinder kept at the IBWM at Sevres, France.					The second Initially the second was defined at 1/60 * 1/60 * 1/24 of a mean solar day. We know that the earth's rotation is slowing, so now the second is defined by 9192631770 vibrations of a Cs atom.
Demo: Standard Units of Measurement: ME-A-SU Demo: Volume Comparison: ME-A-VC cc’s from 1 g/cc from meters
1.4 Dimensional Analysis & 1.5 Conversion of Units
You are driving along the highway with you speed of 60 mph So the units of speed is miles / hour · Speed à miles / hour · Speed à meters / second * New Formula * speed = distance / time So is it possible for the units of speed to be m/s^2? m / s ≠ m / s^2 So this is an impossibility		You are at a red traffic light. The light turns green. What do you do? Ans: You step on the accelerator (or gas pedal). See table to the right à If you notice the Δspeed per unit time is constant Δ is read as “change of” * New Formula * acceleration = Δ velocity / Δtime						The accelerator increases speed per unit time. 0 to 1 sec: 0 to 5 mph 1 to 2 sec: 5 to 10 mph 2 to 3 sec: 10 to 15 mph 3 to 4 sec: 15 to 20 mph 4 to 5 sec: 20 to 25 mph In lecture we discuss the difference between speed (a magnitude) and speed with direction (magnitude w/direction). Velocity is speed with direction which is a vector quantity
You are driving along the highway with you speed of 60 mph. Let’s convert this to miles per hour using estimations. 60 miles 1 hour 1800 yard 1 meter = 30 m/s or 2 mph ≈ 1 m/s hour 3600 sec 1 mile 1 yard As you can see…these are rough approximations. In reality there is about 2.2 mph = 1.0 m/s But the IMPORTANT key to recognize…these calculations are easily completed…no calculator is needed !
1.7 Uncertainty (Sig Figures from book)
· Always reported as absolute uncertainty · NEVER show fractional uncertainty in a decimal form (looks like absolute uncertainty) · Rules below w/ examples · Always carry and extra sig fig through work until the final answer
a = 10 ± 1	b = 2.0 ± 0.3						c = 3.0 ± 0.1
Uncertainty that is added or subtracted: ABSOLUTE uncertainty is added
a + b = (10 ± 1) + (2.0 ± 0.3) = 10 + 2.0   ± (1 + .3) = 12 ± 1.3 a + b = 12 ± 1 Or 12 ± 2 (uncertainties always become larger with rounding)					a – b = (10 ± 1) - (2.0 ± 0.3) = 10 – 2.0    ± (1 + .3) = 8.0 ± 1.3 a - b = 8 ± 1 Or 8 ± 2
Uncertainty that is multiplied or divided: FRACTIONAL (or percent) uncertainty is added At this point we must convert the absolute uncertainty to Fractional
a * b = (10 ± 1)      * (2.0 ± 0.3) = (10 ± 1/10) * (2.0 ± 0.3/2) = 10 * 2 ± (1/10 + 0.3/2) = 20 ± (10% + 15%) = 20 ± 25% = 20 ± 25% * 20 -------> Converting back to ABSOLUTE = 20 ± 5					a / b = (10 ± 1) / (2.0 ± 0.3) = (10 ± 1/10) / (2.0 ± 0.3/2) = (10 ± 10%) / (2.0 ± 15%) = 10 / 2 ± (10% + 15%) = 5 ± 25% = 5 ± 25% * 5 -------> Converting back to ABSOLUTE = 5 ± 1.25 (Follow one sig fig rule) = 5 ± 1 OR 5 ± 2 (Uncertainties round up)
Uncertainty that is raised to a power: FRACTIONAL (or percent) uncertainty is multiplied by that power
a^2 = (10 ± 1)^2 = (10 ± 1/10)^2 = (10 ± 10%)^2 = 10^2 ± 10% * 2 = 100 ± 20% = 100 ± (20% * 100) = 100 ± 20					a^1/2 = (10 ± 1)^½ = (10 ± 1/10)^½ = (10 ± 10%)^½ = 10^½ ± (10% * ½) = 3.2 ± 5% = 3.2 ± (5% * 3.2) = 3.2 ± 0.2
a * b + c = (10 ± 1)      * (2.0 ± 0.3) + 3.0 ± 0.1 = (10 ± 1/10) * (2.0 ± 0.3/2) + 3.0 ± 0.1 = (10 ± 10%) * (2.0 ± 15%) + 3.0 ± 0.1 = 20 ± 25% + 3.0 ± 0.1 = 20 ± (25% * 20) + 3.0 ± 0.1 = 20 ± 5 + 3.0 ± 0.1 = 20 + 3 ± (5 + 0.1) = 23 ± 5.1 (but uncertainty must be to same decimal place) = 23 ± 6					a / b - c = (10 ± 1)      / (2.0 ± 0.3) - 3.0 ± 0.1 = (10 ± 1/10) / (2.0 ± 0.3/2) - 3.0 ± 0.1 = (10 ± 10%) / (2.0 ± 15%) - 3.0 ± 0.1 = 5 ± 25% - 3.0 ± 0.1 = 5 ± (25% * 5) - 3.0 ± 0.1 = 5 ± 1.25 - 3.0 ± 0.1 = 5 – 3.0 ± (1.25 + 0.1) = 2 ± 1.35 = 2 ± 2 (uncertainty should not round down) or 2.0 ± 1.4 (I prefer…but doesn’t follow 1 sig fig for uncertainty rule)
a^3 * b / c = (10 ± 1)^3      * (2.0 ± 0.3) / 3.0 ± 0.1 = (1000 ± 3(1/10)) * (2.0 ± 0.3/2) / 3.0 ± 0.1/3 = [1000 ± 3(10%)] * (2.0 ± 15%) / 3.0 ± 3.33% = 1000 * 2 / 3 ± (3*10% + 15% + 3.33%) = 667 ± 48.3% = 667 ± (48.3% * 667) = 667 ± 322 = 670 ± 330 or Remember…I prefer 2 sig figs for uncertainty…but rule states = 700 ± 400 you can only have one sig fig for uncertainty
One last uncertainty problem a^3 * b / c - (a – b) = (10 ± 1)^3      * (2.0 ± 0.3) / 3.0 ± 0.1 - [10 ± 1) - (2.0 ± 0.3] = (1000 ± 3(1/10)) * (2.0 ± 0.3/2) / 3.0 ± 0.1/3 - [10 - 2    ± (1 + .3)] = (1000 ± 3(10%)) * (2.0 ± 15%) / 3.0 ± 3.33% - (8 ± 1.3) = 1000 * 2 / 3 ± (30% + 15% + 3.33%) - (8 ± 1.3) = 667 ± 48.3% - (8 ± 1.3) = 667 ± 48.3% * 667 - (8 ± 1.3) = 667 ± 322 - (8 ± 1.3) = (667 – 8) ± (322 + 1.3) = 659 ± 323 = 660 ± 330 (I prefer this) = 700 ± 400 (Standard recognized correct answer) So the real value falls somewhere between 300 to 1100 (quite a big range)
Another uncertainty problem 4 / a^(1/2)* ( a * b - a * c / b^2) = 4 /(10 ± 1)^1/2 * [10 ± 1 * 2.0 ± 0.3 - 10 ± 1 * 3.0 ± 0.1 / (2.0 ± 0.3)^2] = 4/3.32 ± (½ 1/10) * [20 ± (1/10 + 0.3/2.0) - 7.5 ± (1/10 + 0.1/3 + 2 (0.3/2) ] = 1.20 ± (½ 10%) * [20 ± (10% + 15%) - 7.5 ± (10% + 3.33% + 2*15%) ] = 1.20 ± 5% * [20 ± 25% - 7.5 ± 43.33% ] = 1.20 ± 5% * [20 ± 5 - 7.5 ± 3.25 ] = 1.20 ± 5% * (12.5 ± 8.25) = 1.20 ± 5% * (12.5 ± 8.25/12.5) = 1.20 ± 5% * (12.5 ± 66%) = 15 ± 71% = 15 ± 10.65 = 15 ± 11 (Using 2 sig figs for uncertainty)
1.7 (121) Scalars & Vectors
One day you decide to drive your car for 1 hour. There is no traffic and you drive at a speed of 65 miles per hour. Where are you? You can easily determine you have traveled about 65 miles. But which direction?						SCALAR: 65 miles is a scalar. This is just a magnitude of a value, no direction is given. You must also know which direction to know you final location. VECTOR: A magnitude WITH a direction is called a vector.