University Physics Ch 01: Units, Physical Quantities & Vectors Practice Flashcards

Algebra Review: Simplifying Expressions and Exponents

• Simplifying Algebraic Expressions: The objective is to write long algebraic expressions in a simpler way by reducing the number of terms.

  • Step 1: Distribute constants or variables into parentheses (if any).

  • Step 2: Group like terms by writing them next to each other.

  • Step 3: Combine like terms by adding or subtracting.

  • Example: $2x + 3 + 4(x + 2)$. Distributing: $2x + 3 + 4x + 8$. Combining: $6x + 11$.

  • Example: $4x + 6 - 3(x + 2)$. Distributing: $4x + 6 - 3x - 6$. Combining: $x$.

• Exponents in Expressions: Exponents represent repeated multiplication of a base.

  • Base: The number and/or variable being multiplied.

  • Exponent (Power): The indicator of how many times the base is multiplied.

  • General Form: $a^n = a \times a \times \dots \times a$ ($n$ times).

  • Example: $4^5 = 4 \times 4 \times 4 \times 4 \times 4$ ("4 to the 5th power").

• Exponent Rules:

  • Product Rule: Multiply same bases by adding exponents: $a^m \times a^n = a^{m+n}$. Example: $4^3 \times 4^2 = 4^5$.

  • Quotient Rule: Divide same bases by subtracting exponents: $\frac{a^m}{a^n} = a^{m-n}$. Example: $\frac{4^5}{4^2} = 4^3$.

  • Zero Exponent Rule: Anything to the zero power equals 1: $a^0 = 1$. Example: $\frac{4^3}{4^3} = 4^0 = 1$.

  • Negative Exponent Rule: A negative exponent in the numerator moves to the bottom with a positive exponent, and vice versa: $a^{-n} = \frac{1}{a^n}$ or $\frac{1}{a^{-n}} = a^n$. Example: $\frac{4^2}{4^5} = 4^{-3} = \frac{1}{4^3}$.

  • Power Rule: Power to another power requires multiplying exponents: $(a^m)^n = a^{m \times n}$. Example: $(4^2)^3 = 4^6$.

  • Power of a Product: Distribute the exponent to each term in the parentheses: $(a \cdot b)^n = a^n \cdot b^n$. Example: $(3 \cdot 4)^2$.

  • Power of a Quotient: Distribute the exponent to both numerator and denominator: $(\frac{a}{b})^n = \frac{a^n}{b^n}$. Example: $(\frac{12}{4})^2$.

Equations and Coordinate Graphing

• Solving Linear Equations: Using operations ($+$, $-$, $\times$, $\div$) to isolate the variable $x$. Always perform operations on both sides of the equation.

  • Step 1: Distribute constants.

  • Step 2: Combine like terms.

  • Step 3: Group terms with $x$ on one side and constants on the other.

  • Step 4: Isolate and solve for $x$.

  • Step 5: Check the solution by substituting back into the original equation.

  • Example 1: $2(x - 3) = 0 \rightarrow 2x - 6 = 0 \rightarrow 2x = 6 \rightarrow x = 3$.

  • Example 2: Solve $\frac{3}{5} + x + 5 = -3$.

• Solving Systems of Equations by Substitution: Find solutions $(x, y)$ that satisfy multiple equations.

  • Step 1: Solve one equation (A) for terms of one variable (e.g., $y$).

  • Step 2: Plug the resulting expression into the second equation (B).

  • Step 3: Solve equation (B) for the remaining variable.

  • Step 4: Plug that value back into equation (A) to find the first variable.

  • Example: Solve $y = 3x - 6$ (A) and $2x + y = 4$ (B). Substitute A into B: $2x + (3x - 6) = 4 \rightarrow 5x = 10 \rightarrow x = 2$. Then solve for $y$: $y = 3(2) - 6 = 0$. Solution: $(2, 0)$.

• Graphing Equations by Plotting Points:

  • Step 1: Isolate $y$ on the left side ($y = f(x)$).

  • Step 2: Calculate $y$-values from 3-5 chosen $x$-values.

  • Step 3: Plot the resulting $(x, y)$ ordered pairs on the 2D rectangular coordinate system.

  • Step 4: Connect the points with a line or curve.

  • Example: Graph $y = x^2 - 3x + 2$.

    • If $x=0$, $y=2$. Ordered pair: $(0, 2)$.

    • If $x=1$, $y=0$. Ordered pair: $(1, 0)$.

    • If $x=2$, $y=0$. Ordered pair: $(2, 0)$.

    • If $x=3$, $y=2$. Ordered pair: $(3, 2)$.

    • If $x=4$, $y=6$. Ordered pair: $(4, 6)$.

Linear Equations and Slopes

• Slope ($m$): A number representing the steepness of a line; defined as the change in $y$ ($\Delta y$) divided by the change in $x$ ($\Delta x$).

  • Formula: $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$.

  • Example Line A with points $(1, 2)$ and $(2, 4)$: $\Delta y = 4 - 2 = 2$, $\Delta x = 2 - 1 = 1$, $m = \frac{2}{1} = 2$.

• Slope-Intercept Form: $y = mx + b$.

  • $m$: Slope.

  • $b$: $y$-intercept.

  • Steps to Graph: 1) Plot $y$-intercept $(0, b)$. 2) Plot the next point using the slope. 3) Connect points.

  • Example: $y = 2x - 1$. Here, $b = -1$ and $m = 2$.

Quadratic Equations: Solving and Graphing

• Standard Form of a Quadratic: $ax^2 + bx + c = 0$.

• Square Root Property:

  • Use if the equation follows the form $(x + k)^2 = \text{constant}$ or if there is no middle term ($b = 0$).

  • Steps: 1) Isolate the squared expression. 2) Take the plus and minus square root. 3) Solve for $x$. 4) Check solutions.

  • Example: $(x + 1)^2 = 4 \rightarrow x + 1 = \pm 2 \rightarrow x = 1$ or $x = -3$.

• Quadratic Formula:

  • Use if the equation is hard to factor or if you are unsure.

  • Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

  • Example: $x^2 + 2x - 3 = 0$. Here $a=1$, $b=2$, $c=-3$.

• Graphing Quadratic Equations:

  • Vertex Form: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

  • Steps: 1) Identify the vertex (minimum or maximum). 2) Find $x$-intercepts by solving $y=0$. 3) Find $y$-intercept by setting $x=0$. 4) Identify ranges of increasing and decreasing behaviors. 5) Connect with a smooth curve.

  • Example: $y = -(x - 1)^2 + 4$ (Vertex at $(1, 4)$, downward parabola).

Proportional Reasoning

• Directly Proportional: $y = kx$. As $x$ increases ($\uparrow$), $y$ increases ($\uparrow$). As $x$ decreases ($\downarrow$), $y$ decreases ($\downarrow$).

• Inversely Proportional: $y = \frac{k}{x}$. As $x$ increases ($\uparrow$), $y$ decreases ($\downarrow$).

• Jointly Proportional: $F = m \cdot a$. Variables change relative to multiple others. For example, if $F$ is constant, if $m$ increases ($\uparrow$), $a$ must decrease ($\downarrow$).

Trigonometry Review

• Right Triangle Relationships (SOH CAH TOA):

  • Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. Rearranged: $\text{opp} = \text{hyp} \cdot \sin(\theta)$.

  • Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. Rearranged: $\text{adj} = \text{hyp} \cdot \cos(\theta)$.

  • Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Also: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

• Pythagorean Theorem: For a right triangle with sides $a$, $b$ and hypotenuse $c$: $a^2 + b^2 = c^2$.

• Trigonometric Identities: $\sin^2(\theta) + \cos^2(\theta) = 1$.

Calculus Essentials

• Derivatives: Represents the instantaneous rate of change or the slope of a tangent line to a curve.

  • Constant Rule: If $f(x) = c$, then $f'(x) = 0$.

  • Linear Rule: If $f(x) = cx$, then $f'(x) = c$.

  • Power Rule: If $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$.

  • Sum Rule: If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$.

  • Examples:

    • $f(x) = 3 \rightarrow f'(x) = 0$.

    • $f(x) = -2x \rightarrow f'(x) = -2$.

    • $f(x) = x^2 \rightarrow f'(x) = 2x$.

    • $f(x) = x^2 + 3x \rightarrow f'(x) = 2x + 3$.

• Integrals: Graphically represents the area under the curve. They are the reverse of derivatives.

  • Constant Rule: $\int c \,dx = cx + C$.

  • Power Rule ($n \neq -1$): $\int x^n \,dx = \frac{1}{n+1}x^{n+1} + C$.

  • Constant Multiple Rule: $\int cx^n \,dx = c \cdot \int x^n \,dx = c \cdot \frac{1}{n+1}x^{n+1} + C$.

  • Definite Integral: The integral from point $a$ to $b$: $\int_a^b f(x) \,dx = F(b) - F(a)$.

  • Examples:

    • $\int 3 \,dx = 3x + C$.

    • $\int x^2 \,dx = \frac{1}{3}x^3 + C$.

    • $\int -2x \,dx = -x^2 + C$.

Units and the S.I. System

• Physical Quantities: Measurements consist of a Number and a Unit. Physics equations only work if units are compatible.

• S.I. Units (Système International):

  • Mass: Kilogram ($kg$). [Imperial: Pound ($lb$)]

  • Length: Meter ($m$). [Imperial: Foot ($ft$)]

  • Time: Second ($s$). [Imperial: Second ($s$)]

  • Force: Newton ($N$). [Imperial: Foot-pound]

• Compatibility Check: In the formula $F = m \times a$, units must match ($N = kg \cdot m/s^2$).

Metric Prefixes

• Metric Prefixes Table:

  • Tera (T): $10^{12}$

  • Giga (G): $10^9$

  • Mega (M): $10^6$

  • Kilo (k): $10^3$

  • Hecto (h): $10^2$

  • Deca (da): $10^1$

  • [Base Unit]: $10^0 = 1$

  • Deci (d): $10^{-1}$

  • Centi (c): $10^{-2}$

  • Milli (m): $10^{-3}$

  • Micro ($\mu$): $10^{-6}$

  • Nano (n): $10^{-9}$

  • Pico (p): $10^{-12}$

• Unit Shifting Rule:

  • Moving from a bigger unit to a smaller unit: The number becomes Larger.

  • Moving from a smaller unit to a bigger unit: The number becomes Smaller.

• Steps for Prefix Conversion:

  1. Identify starting and target prefixes.

  2. Count the number of exponents moved from start to target.

  3. Shift the decimal place in the same direction moved in Step 2.

• Prefix Examples:

  • $5\,km = 5 \times 10^3\,m = 5000\,m$.

  • $4.6\,ms = 4.6 \times 10^{-3}\,s = 0.0046\,s$.

  • $6.5\,hm$ to $m = 650\,m$.

  • $3.89\,mm$ to $m = 0.00389\,m$.

  • $7.62\,kg$ to $\mu g = 7.62 \times 10^9\,\mu g$.

  • Practice: Earth's circumference is $40.1\,Mm$. In kilometers? $\text{Target: km}$. Move from Mega ($10^6$) to Kilo ($10^3$) is 3 places right. Answer: $40,100\,km$.

  • Practice: Wavelength of $3,000,000,000\,nm$ in decameters ($dam$)? Move from Nano ($10^{-9}$) to Deca ($10^1$) is 10 places left. Answer: $0.3\,dam$.

Scientific Notation

• Purpose: To condense very long or inconvenient numbers into shorter ones.

• General Format: $A.BC \times 10^D$, where 1 \le |A.BC| < 10.

• Standard Form to Scientific Notation:

  1. Move the decimal point until the leading number is $\ge 1$ but < 10.

  2. Round to 2 decimal places if needed.

  3. The number of places moved is the exponent ($D$). If the original number was > 10, $D$ is positive. If < 1, $D$ is negative.

  • Example: Mass of Earth $= 5,972,000,000,000,000,000,000,000\,kg = 5.97 \times 10^{24}\,kg$.

  • Example: $304,605.27\,kg \rightarrow 3.05 \times 10^5\,kg$.

  • Example: $0.000102\,m \rightarrow 1.02 \times 10^{-4}\,m$.

• Scientific Notation to Standard Form:

  1. The exponent indicates the number of decimal places to move.

  2. If positive, move right (number grows larger). If negative, move left (number grows smaller).

  • Example: $5.45 \times 10^2\,kg = 545\,kg$.

  • Example: $9.62 \times 10^{-5}\,s = 0.0000962\,s$.

Unit Conversions

• Importance: Non-S.I. units must be converted to S.I. units before plugging them into physics equations.

• Common Conversion Factors:

  • Mass: $1\,kg = 2.2\,lbs$, $1\,lb = 450\,g$, $1\,oz = 28.4\,g$.

  • Length: $1\,km = 0.621\,mi$, $1\,ft = 0.305\,m$, $1\,in = 2.54\,cm$.

  • Volume: $1\,gal = 3.79\,dm^3$, $1\,cm^3 = 1\,mL$, $1\,dm^3 = 1.06\,qt$.

• Conversion Steps:

  1. Write down the given unit and the target unit.

  2. Write conversion factors as fractions (ratios) so that initial units are in the denominator and cancel out.

  3. Multiply all numerators and divide by the product of all denominators.

  • Example: Convert $22\,lbs$ to $kg$: $22\,lbs \times \frac{1\,kg}{2.2\,lbs} = 10\,kg$.

  • Example with exponents: $100\,ft^2$ to $m^2$: $100\,ft^2 \times (\frac{0.305\,m}{1\,ft}) \times (\frac{0.305\,m}{1\,ft}) = 9.30\,m^2$.

  • Example: $67.5\,mi/hr$ to $m/s$: $67.5\,mi/hr \times \frac{1609\,m}{1\,mi} \times \frac{1\,hr}{3600\,s} \approx 30.2\,m/s$.

Density and Geometry

• Density (\rho): Defined as mass divided by volume.

  • Formula: $\rho = \frac{m}{V}$. Units: $kg/m^3$.

• Volume Formulas for Geometric Shapes:

  • Rectangular Prism: $V = l \times w \times h$.

  • Sphere: $V = \frac{4}{3}\pi R^3$.

  • Cylinder: $V = \pi R^2h$.

• Example Problems:

  • Earth's Mass: Average density $\rho = 5500\,kg/m^3$. Assume sphere radius $R = 3960\,mi$. First, convert $mi$ to $m$: $3960\,mi \times 1609\,m/mi = 6.37 \times 10^6\,m$. Volume $V = \frac{4}{3}\pi (6.37 \times 10^6)^3 = 1.08 \times 10^{21}\,m^3$. Mass $m = \rho \times V = 5500 \times 1.08 \times 10^{21} = 5.94 \times 10^{24}\,kg$.

  • Wooden Cylinder: Radius $3.5\,cm$, height $6\,cm$, mass $161\,g$. Density? $V = \pi (3.5)^2(6) = 230.9\,cm^3$. $\rho = 161/230.9 = 0.697\,g/cm^3$. Converting to $kg/m^3 \rightarrow 697\,kg/m^3$.

Dimensional Analysis

• Dimensional Consistency: Units on the left side of an equation must equal the final units on the right side. This allows the checking of equation validity without calculations.

• Consistency Steps:

  1. Replace every variable with its units.

  2. Ignore constants (fractions like $1/2$, numbers like $2$) and negative signs.

  3. Multiply and divide to cancel units.

  4. Confirm if the left side equals the right side.

• Determining Units of New Variables:

  • Example: Hooke's Law $F = -kx$. Force $F$ in Newtons ($N = kg \cdot m/s^2$), distance $x$ in meters ($m$). Solving for units of $k$: $k = \frac{F}{x} = \frac{kg \cdot m/s^2}{m} = kg/s^2$.

  • Example: Newton's Law of Gravitation $F = G \frac{m_1 m_2}{r^2}$. Units of $G$? $G = \frac{F \cdot r^2}{m_1 m_2} = \frac{(kg \cdot m/s^2) \cdot m^2}{kg \cdot kg} = \frac{m^3}{kg \cdot s^2}$.

Significant Figures (Sig Figs)

• Precision and Significance: Measurements in physics have precision. Not all digits are significant. Sig figs are the digits that matter for precision.

• Counting Rules:

  1. Eliminate Leading Zeros: Zeros at the start ($0.00...$) never count.

  2. Eliminate Trailing Zeros (If no decimal): In numbers like $100$, zeros are placeholders. If there is a decimal (e.g., $100.0$), they count.

  3. Always Count: Non-zero digits and middle zeros (sandwiched zeros).

• Examples:

  • $100.00$: 5 Sig Figs.

  • $0.0043$: 2 Sig Figs.

  • $31000092$: 8 Sig Figs.

  • $100$: 1 Sig Fig.

  • $73917000$: 5 Sig Figs.

  • $0.00900$: 3 Sig Figs.

  • $0.0132009720000$: 9 Sig Figs (leading zeros removed: $132009720000$ is wrong; should be $132009720...$ wait, following rule: leading zeros removed, then count). Correct: $132009720000$ is 12 digits.

• Math with Sig Figs:

  • Rule 1 (Addition/Subtraction): Round the answer to the same number of Decimal Places as the value with the least decimal places.

  • Rule 2 (Multiplication/Division): Round the answer to the same number of Sig Figs as the number with the least sig figs.

  • Rule 3 (Mixed Operations): Follow PEMDAS. Keep extra digits in intermediate steps, marking the last significant digit. Round only the final answer.

  • Example: Area of sidewalk $2.293\,m$ wide and $90\,m$ long. $2.293 \times 90 = 206.37$. Round to 1 Sig Fig (due to "90"): $200\,m^2$. Note: If "90" is exactly $90$, it might have more sig figs depending on context, but strictly, $200\,m^2$ follows the rule.

Vectors and Scalars

• Scalar: A measurement with magnitude (size) only. Examples: Mass ($5\,kg$), Time ($24\,hr$), Temperature ($60^{\circ}F$), Distance ($10\,ft$), Speed ($80\,mph$).

• Vector: A measurement with both magnitude and direction. Examples: Force ($100\,N$ left), Displacement ($10\,ft$ East), Velocity ($80\,mph$ West).

• Displacement (\vec{\Delta x}) vs. Distance (d):

  • Distance: Scalar representing the total length of the path traveled. Always positive.

  • Displacement: Vector representing the change in position ($x_{final} - x_{initial}$). Can be negative (indicating direction).

Vector Math Fundamentals

• Graphical Addition: Add vectors "tip-to-tail". The Resultant Vector (R) is the shortest path from the start of the first vector to the end of the last.

• Parallel Vectors: Add like normal numbers (e.g., $3\,m$ right + $4\,m$ right $= 7\,m$ right).

• Perpendicular Vectors: Use Triangle Math (Pythagorean theorem).

  • Example: Walk $3\,m$ right, then $4\,m$ up. Total displacement magnitude $= \sqrt{3^2 + 4^2} = 5\,m$.

• Subtracting Vectors: Adding the negative of a vector. A negative vector has the same magnitude but points in the opposite direction.

• Multiples of Vectors: Multiplying a vector by a scalar changes magnitude but not direction (unless the scalar is negative).

Vector Composition and Decomposition

• Vector Decomposition (Vector \rightarrow Components): Breaking a vector into its vertical ($y$) and horizontal ($x$) parts using trigonometry.

  • $A_x = A \cos(\theta_x)$.

  • $A_y = A \sin(\theta_x)$.

• Vector Composition (Components \rightarrow Vector): Combining components to find the full vector.

  • Magnitude: $A = \sqrt{A_x^2 + A_y^2}$.

  • Direction (Reference Angle): $\theta_x = \tan^{-1}(\frac{|A_y|}{|A_x|})$.

Vector Addition by Components

• Steps to add multiple vectors algebraically:

  1. Draw and connect vectors tip-to-tail.

  2. Calculate all $x$ and $y$ components for each vector individually.

  3. Sum the $x$-components to get $R_x$. Sum the $y$-components to get $R_y$.

  4. Calculate the resultant magnitude: $R = \sqrt{R_x^2 + R_y^2}$.

  5. Calculate the resultant angle: $\theta_R = \tan^{-1}(\frac{|R_y|}{|R_x|})$.

• Vector Signs in Quadrants:

  • Quadrant 1 ($+x, +y$): Angle $0^{\circ}$ to $90^{\circ}$.

  • Quadrant 2 ($-x, +y$): Angle $90^{\circ}$ to $180^{\circ}$.

  • Quadrant 3 ($-x, -y$): Angle $180^{\circ}$ to $270^{\circ}$.

  • Quadrant 4 ($+x, -y$): Angle $270^{\circ}$ to $360^{\circ}$.

• Direction Descriptions:

  • Counterclockwise (CCW): Positive angle ($+$).

  • Clockwise (CW): Negative angle ($-$).

  • 30^{\circ} North of East: Start East, curve $30^{\circ}$ toward North.

Unit Vectors

• Definition: Special vectors with a magnitude of 1 used to indicate direction.

  • $\hat{i}$: Points in the $+x$ direction.

  • $\hat{j}$: Points in the $+y$ direction.

  • $\hat{k}$: Points in the $+z$ direction.

• Notation: $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$.

• Addition with Unit Vectors: Simply add the corresponding components.

  • Example: $\vec{A} = 4\hat{i} + 2\hat{j}$ and $\vec{B} = -1\hat{i} + 2\hat{j}$. Resultant $\vec{R} = (4-1)\hat{i} + (2+2)\hat{j} = 3\hat{i} + 4\hat{j}$.

Dot Product (Scalar Product)

• Definition: Multiplication of vectors resulting in a Scalar (number only).

• Geometric Formula: $\vec{A} \cdot \vec{B} = |A| |B| \cos(\theta)$, where $\theta$ is the smallest angle between tails.

• Properties:

  • Multiplies parallel components.

  • Positive if components are in the same direction.

  • Negative if components are in opposite directions.

  • Zero if vectors are perpendicular ($\theta = 90^{\circ}$).

• Component Formula: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$.

• Finding Angle between Vectors: $\cos(\theta) = \frac{A_x B_x + A_y B_y + A_z B_z}{|A| |B|}$.

Vector Product (Cross Product)

• Definition: Multiplication of vectors resulting in a Vector perpendicular to both original vectors.

• Geometric Formula (Magnitude): $|\vec{C}| = |\vec{A} \times \vec{B}| = |A| |B| \sin(\theta)$.

• Direction (Right-Hand-Rule):

  1. Point fingers along the 1st vector (A).

  2. Curl fingers toward the 2nd vector (B).

  3. The thumb points in the direction of the product vector (C).

  • $\odot$: Out of the page.

  • $\otimes$: Into the page.

• Properties:

  • Cross product is Zero if vectors are parallel ($\theta = 0^{\circ}$ or $180^{\circ}$).

  • Direction is always perpendicular to the plane formed by A and B.

• Component Calculation (Determinant Method):

  • $C_x = A_y B_z - A_z B_y$

  • $C_y = A_z B_x - A_x B_z$

  • $C_z = A_x B_y - A_y B_x$

• Steps: Build a table of components ($x, y, z, x, y$) and multiply diagonally (cross multiply).

Density and Geometry

• Density ((\rho)): Defined as mass divided by volume.

  • Formula: (\rho = \frac{m}{V}). Units: (kg/m^3).

• Volume Formulas for Geometric Shapes:

  • Rectangular Prism: (V = l \times w \times h).

  • Sphere: (V = \frac{4}{3}\pi R^3).

  • Cylinder: (V = \pi R^2h).

• Example Problems:

  • Earth's Mass: Average density (\rho = 5500\,kg/m^3). Assume sphere radius (R = 3960\,mi). First, convert (mi) to (m): (3960\,mi \times 1609\,m/mi = 6.37 \times 10^6\,m). Volume (V = \frac{4}{3}\pi (6.37 \times 10^6)^3 = 1.08 \times 10^{21}\,m^3). Mass (m = \rho \times V = 5500 \times 1.08 \times 10^{21} = 5.94 \times 10^{24}\,kg).

  • Wooden Cylinder: Radius (3.5\,cm), height (6\,cm), mass (161\,g). Density? (V = \pi (3.5)^2(6) = 230.9\,cm^3). (\rho = 161/230.9 = 0.697\,g/cm^3). Converting to (kg/m^3 \rightarrow 697\,kg/m^3).

Dimensional Analysis

• Dimensional Consistency: Units on the left side of an equation must equal the final units on the right side. This allows the checking of equation validity without calculations.

• Consistency Steps:

  1. Replace every variable with its units.

  2. Ignore constants (fractions like (1/2), numbers like (2)) and negative signs.

  3. Multiply and divide to cancel units.

  4. Confirm if the left side equals the right side.

• Determining Units of New Variables:

  • Example: Hooke's Law (F = -kx). Force (F) in Newtons ((N = kg \cdot m/s^2)), distance (x) in meters ((m)). Solving for units of (k): (k = \frac{F}{x} = \frac{kg \cdot m/s^2}{m} = kg/s^2).

  • Example: Newton's Law of Gravitation (F = G \frac{m_1 m_2}{r^2}). Units of (G)? (G = \frac{F \cdot r^2}{m_1 m_2} = \frac{(kg \cdot m/s^2) \cdot m^2}{kg \cdot kg} = \frac{m^3}{kg \cdot s^2}).

Significant Figures (Sig Figs)

• Precision and Significance: Measurements in physics have precision. Not all digits are significant. Sig figs are the digits that matter for precision.

• Counting Rules:

  1. Eliminate Leading Zeros: Zeros at the start ((0.00…)) never count.

  2. Eliminate Trailing Zeros (If no decimal): In numbers like (100), zeros are placeholders. If there is a decimal (e.g., (100.0)), they count.

  3. Always Count: Non-zero digits and middle zeros (sandwiched zeros).

• Examples:

  • (100.00): 5 Sig Figs.

  • (0.0043): 2 Sig Figs.

  • (31000092): 8 Sig Figs.

  • (100): 1 Sig Fig.

  • (73917000): 5 Sig Figs.

  • (0.00900): 3 Sig Figs.

  • (0.0132009720000): 9 Sig Figs.

• Math with Sig Figs:

  • Rule 1 (Addition/Subtraction): Round the answer to the same number of Decimal Places as the value with the least decimal places.

  • Rule 2 (Multiplication/Division): Round the answer to the same number of Sig Figs as the number with the least sig figs.

  • Rule 3 (Mixed Operations): Follow PEMDAS. Keep extra digits in intermediate steps, marking the last significant digit. Round only the final answer.

  • Example: Area of sidewalk (2.293\,m) wide and (90\,m) long. (2.293 \times 90 = 206.37). Round to 1 Sig Fig: (200\,m^2).