1.1 Measurements in Physics

Fundamental Units

Quantity SI Unit Symbol
Lenght meter m
Mass kilogram kg
Time second s
Current ampere A
Temperature Kelvin K
Amount of Substance mole mol
Luminous intensity candela cd

Rules for significant digits

1.Every non zero digit is significant

2.Zeros between two significant digits are significant

3.Zeros to the right of a significant digit and a decimal point are significant

Metric multipliers (Prefixes)

Factor Name Symbol
1015 Peta P
1012 Tera T
109 Giga G
106 Mega M
103 Kilo k
102 Hecto h
101 Deca da
10-3 milli m
10-6 micro µ
10-9 nano n
10-12 pico p

Metric conversion

1. Simple conversion (fundamental unit to a prefix)

\begin{equation*} \left(\frac{given\ value\ \ given\ unit}{1}\right)\left(\frac{desired\ unit}{undesired\ unit}\right) \end{equation*}

Example: 590.1g -> ng

\begin{equation} \left(\frac{590.1g}{1}\right)\left(\frac{1ng}{10^{-9} g}\right) =590.1*10^{9} ng \end{equation}

2. Double conversion (prefix to a prefix)

\begin{equation} \left(\frac{given\ value\ \ given\ unit}{1}\right)\left(\frac{desired\ unit}{undesired\ unit}\right)\left(\frac{desired\ prefix}{undesired\ prefix}\right) \end{equation}

Example: 706.0 Mm -> nm

\begin{equation} \left(\frac{706.0Mm}{1}\right)\left(\frac{10^{6} m}{1Mm}\right)\left(\frac{1nm}{10^{-9} m}\right) =7.060*10^{17} nm \end{equation}

3. Complex conversion (trick - denominator)

\begin{equation} {\textstyle \left(\frac{given\ value\ \ given\ unit}{1}\right)\left(\frac{desired\ unit}{undesired\ unit}\right)\left(\frac{desired\ prefix}{undesired\ prefix}\right)\left(\frac{undesired\ time}{desired\ time}\right)} \end{equation}

Example: 0.00705µg/s -> dag/hr

\begin{equation} {\textstyle \left(\frac{0.00705µ g}{1}\right)\left(\frac{10^{-6} g}{1µ g}\right)\left(\frac{1dag}{10^{1} g}\right)\left(\frac{3600s}{h}\right)} \end{equation}

Accuracy and precision

Precision in experiments

Precision for analog instruments: half the least increment

Precision for digital instruments: to the last measured digit

1.2 Uncertainties and errors

\begin{gather*} absolute\ uncertainty\ =\ \Delta x\\ fractional\ uncertainty\ =\ \frac{\Delta x}{x}\\ percentage\ uncertainty\ =\ \frac{\Delta x}{x} \ *\ 100\% \end{gather*}
\begin{gather*} y=a\pm b,\ \Delta y=\Delta a+\Delta b\\ y=\frac{a*b}{c} ,\ \frac{\Delta y}{y} =\frac{\Delta a}{a} +\frac{\Delta b}{b} +\frac{\Delta c}{c}\\ y=a^{n} ,\ \frac{\Delta y}{y} =\frac{n\Delta a}{a} \end{gather*}
*Never subtract uncertainties!

Random and systematic errors

Random Error Systematic Error
-Fluctuations in measurements centered around the true value
-Reduced by taking the average
-Affects precision
-Human error
-Inconsistent room temperature
-Fixed shift in measurement away from the true value
-Affects accuracy
-Zero offset error
-Bad method of measurement
-Incorrect callibration

3 Ways of calculating uncertainty of processed data

(1) Range method
(2) Use the equations
(3) Percentage uncertainty

Uncertainties in graphs

-Linear best fit line: max and min lines
-Non-linear best fit line: linearization
-Error bars: horizontal or vertical, lenght of two absolute uncertainties
-Types of graphs

1.3 Vectors and Scalars

Vectors and scalars

Vector Scalar
-Magnitude and Direction

Vector operations

(1) Scalar operation
\begin{gather*} \vec{A} =\ 4.00\ cm\\ 3\vec{A} \ =\ 12.00\ cm\\ \frac{1}{2}\vec{A} \ =\ 2.00\ cm \end{gather*}
(2) Addition and subtraction
(3) Dot product
\begin{equation*} a\cdot b=\parallel a\parallel \parallel b\parallel cosθ \end{equation*}
(4) Cross product
\begin{equation*} a\times b=\begin{vmatrix} i & j & k\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3} \end{vmatrix} \end{equation*}

Resolving vectors

Topic 1 Problems

1. Big chungus ran 9.2 meters in 3.456 seconds. What is his speed?

A. 2.662 m/s

B. 2.66 m/s

C. 2.7 m/s

D. 3 m/s

2. Which one is not a fundamental quantity?

A. Power

B. Lenght

C. Mass

D. Luminous intensity

3. What is A+B if A = 20.0 feet 30° NE, and B = 30.0 feet North?

A. 44.59ft [67° NE]

B. 23.6ft [43° SW]

C. 37.8fr [56° NW]

D. 43.6ft [66.6° NE]

4. Convert 9870 GJ/s to µJ/h.

A. 3.553 x10-22 µJ/h

B. 3.55 x1022 µJ/h

C. 2.74 x1015 µJ/h

D. 3.553 µJ/h

5. The radius of a baloon is measured to be (20.0±1.0)cm What is the uncertainty in the volume of the ballon? (treat it as sphere)

A. 0.55%

B. 5%

C. 20%

D. 15%

Number of correct answers: