Vectors, 2D Motion, Units, and Dimensional Scaling Notes

Course Administration and Study Resources

• Assignments: Students should begin work on the MasterPlan (MP) assignment immediately. While it is due on Thursday night, it is described as being quite long.

• Support Systems: For physics-related questions, students are encouraged to use the following resources:     * Piazza: For online discussion and queries.     * Visiting Hours: For direct instructor engagement.     * Studio Sessions: Questions can be asked during scheduled studio time.

One-Dimensional Motion Example: Free Fall

• Scenario: A pebble is thrown upward and released from a height of $1.5\,m$ above the ground. It ascends to a maximum height and then falls until it hits the ground.

• Data Provided:     * Release Height ($y_0$): $1.5\,m$     * Total Time ($t$): $1.8\,s$ from release until impact.

• Questions to Solve:     * How high above the ground did the pebble rise at its maximum?     * What was its speed upon impact with the ground?

• Assumptions: Air drag is neglected.

• Calculated Results:     * Maximum Height: The pebble rises $3.25\,m$ above the release point. The total height above the ground is calculated as $1.5\,m + 3.25\,m = 4.75\,m$.     * Impact Velocity ($v$): $-9.65\,m/s$ at the ground.     * Impact Speed: $9.65\,m/s$ (the magnitude of the velocity).

Introduction to Vectors in Two Dimensions

• Definition of Position: The location of a point in space is represented as a vector. Vectors are directional line segments.

• Vector Characteristics:     * Position Vector: An arrow drawn from the origin to the location of the object in space.     * Magnitude: The length of the segment, denoted as $|\vec{r}|$ or simply $r$.     * Direction: The direction in which the arrow points.     * Equality: Two vectors are considered equal if they have the same magnitude (length) and point in the same direction (parallel).

• Vector Notation:     * In 2-D, a vector can be written as: $\vec{r} = x\hat{x} + y\hat{y}$     * Alternatively, it can be written in coordinate form: $\vec{r} = (x, y)$

• Unit Vectors:     * $\hat{x} = (1, 0)$     * $\hat{y} = (0, 1)$

Vector Components and Trigonometry

• Defining Components: Vector components represent how much of the position arrow lies along the $x$ and $y$ axes.

• Trigonometric Relationships: Given an angle $\theta$ (which can often be chosen freely by the user):     * $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{r_y}{|\vec{r}|}$     * $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{r_x}{|\vec{r}|}$     * $r_y = r\sin(\theta)$     * $r_x = r\cos(\theta)$

• Pythagorean Theorem: The magnitude of the vector is related to its components by: $|\vec{r}| = r = \sqrt{r_x^2 + r_y^2}$

• Angle Calculation: The angle can be found using the inverse tangent: $\theta = \tan^{-1}(\frac{r_y}{r_x})$

Vector Addition and Subtraction: Geometrical Method

• Addition ($\vec{A} + \vec{B}$):     * Place the tail of vector $\vec{B}$ onto the head of vector $\vec{A}$.     * The resultant (green vector) connects the tail of $\vec{A}$ to the head of $\vec{B}$.

• Special Cases for Addition:     * Equality (maximum magnitude) occurs if $\vec{A}$ and $\vec{B}$ are in the same direction.     * Minimum magnitude occurs if $\vec{A}$ and $\vec{B}$ are in opposite directions.

• Vector Subtraction: To subtract, add the negative of the vector: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.

• Triangle Inequality: If $\vec{C} = \vec{A} + \vec{B}$, then: $A - B \leq C \leq A + B$

Motion in More Than One Dimension

• Position Vector: The time-dependent location is represented by a 3-dimensional vector $\vec{r}(t)$.

• Velocity ($\vec{v}$): Defined as the derivative of position with respect to time:     * $\vec{v} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$     * For small time intervals: $\Delta \vec{r} = \vec{v}\Delta t$     * Direction: The velocity vector is always aligned with the path of the object.

• Acceleration ($\vec{a}$): Defined as the derivative of velocity with respect to time:     * $\vec{a}(t) = \frac{d\vec{v}}{dt}$     * Direction: Acceleration is aligned with the direction of change in velocity.

Turning Motion and Acceleration Components

• Velocity Update Equation: $\vec{v}_f = \vec{v}_i + \vec{a}\Delta t$

• Tangential Acceleration ($a_{\parallel}$): This is the component of acceleration parallel to the velocity.     * It changes the speed of the object.     * If a_{\parallel} > 0, speed increases.     * If a_{\parallel} < 0, speed decreases.     * Defined as the rate of change of speed: $a_{\parallel} = \frac{dv}{dt}$

• Centripetal Acceleration ($a_{\perp}$): This is the component of acceleration perpendicular to the instantaneous velocity.     * It changes the direction of $\vec{v}$ but does not change the speed.     * The particle turns in the direction of $a_{\perp}$.     * Magnitude: $a_{\perp} = \frac{v^2}{R}$, where $R$ is the radius of curvature at that moment.

Examples: Motion and Acceleration

• Bacterium Movement: A bacterium moves along a curved path with steadily increasing speed at point $P$.     * The total acceleration $\vec{a}$ is the vector sum of $a_{\perp}$ (pointing toward the center of the curve) and $a_{\parallel}$ (pointing along the path of motion in the direction of travel).

• Merry-Go-Round Calculation:     * Data: Diameter = $3.0\,m$ (Radius $R = 1.5\,m$). Acceleration experienced at the edge = $0.5g = 4.9\,m/s^2$.     * Task: Find the time ($T$) for one revolution, assuming constant speed.     * Solution:         1. Use $a_{\perp} = \frac{v^2}{R} \Rightarrow 4.9 = \frac{v^2}{1.5}$.         2. Calculated Speed ($v$) = $2.7\,m/s$.         3. Period ($T$) = $\frac{\text{circumference}}{v} = \frac{2\pi(1.5)}{2.7}$.         4. Calculated Period ($T$) = $3.5\,s$.

Projectile Motion (2D)

• Projectile motion is a combination of two independent behaviors:     1. Horizontal Motion: Constant velocity ($v_x = \text{constant}$, because $a_x = 0$).     2. Vertical Motion: Constant downward acceleration ($a_y = -g$).

• The update equation remains: $\vec{v}_f = \vec{v}_i + \vec{a}\Delta t$

Base Dimensions

• Physical quantities are expressed using a set of base dimensions:     * Length: $[L]$ (Units: meter, mm, cm, etc.)     * Mass: $[M]$ (Units: kg, g, etc.)     * Time: $[T]$ (Units: second, hour, etc.)     * Amount of Substance (mols): $[N]$     * Thermodynamic Temperature: $[\Theta]$     * Current: $[I]$     * Luminous Intensity: $[J]$

Derived Dimensions Examples

• Area: Length $\times$ Length = $[L^2]$

• Volume: Length $\times$ Length $\times$ Length = $[L^3]$

• Speed: Length / Time = $[LT^{-1}]$

• Density: Mass / Volume = $[ML^{-3}]$

Rules of Dimensional Analysis

1. Homogeneity: Every term in an expression must have identical dimensions.

2. Equality: Both sides of an equation must share the same dimensions.     * Example (Consistent): $v_f^2 = v_i^2 + 2a\Delta x$ ($[L^2T^{-2}] = [L^2T^{-2}] + [LT^{-2}][L]$)     * Non-example (Inconsistent): $x = vt^2 + \frac{1}{2}at^3$

3. Dimensionless Functions: Trigonometric functions ($\sin$, $\cos$), exponentials ($e^x$), and logarithms ($\ln$), as well as their arguments, must be dimensionless.     * Example: In $y(x,t) = A\cos(kx - \omega t)$, the argument $(kx - \omega t)$ must be dimensionless.

Units and Scientific Notation

• SI System: The standard system of units (International System of Units).     * Length: meter ($m$)     * Mass: kilogram ($kg$)     * Time: second ($s$)     * Amount: mole ($mol$)     * Temperature: Kelvin ($K$)

• Handling Scale:     * Scientific Notation: Useful for extremely small/large numbers.         * Example: Planck's constant $h = 6.63 \times 10^{-34}\,kg\,m^2/s$.     * Unit Prefixes:         * centi- (c): $10^{-2}$         * milli- (m): $10^{-3}$         * micro- (\mu): $10^{-6}$         * nano- (n): $10^{-9}$         * pico- (p): $10^{-12}$         * kilo- (k): $10^3$         * mega- (M): $10^6$         * giga- (G): $10^9$

Unit Conversion Examples

• Volume Conversion: How many cubic centimeters ($cc$) in a cubic meter?     * ($100\,cm = 1\,m$), so $(100\,cm)^3 = (1\,m)^3 \Rightarrow 10^6\,cm^3 = 1\,m^3$.

• Acre-foot Conversion: Express $0.85$ acre-feet in $m^3$     * Conversion Facts:         * $1\,\text{acre} = 4,840\,\text{yards}^2$         * $1\,\text{yard} = 3\,\text{feet} = 36\,\text{inches}$         * $1\,\text{inch} = 2.54\,cm = 0.0254\,m$

Proportional Reasoning and Scaling

• Definition: $y$ is proportional to $x$ if $y = cx^p$ ($y \propto x^p$).     * Direct/Linear Proportionality ($p = 1$): $y = cx$     * Inverse Proportionality ($p = -1$): $y = c/x$

• Utility: Allows solving problems without exact relations, simplifies math by canceling constants, and provides a check for typos.

• Isomorphic Changes: Occur when all linear length dimensions change by the same factor (same shape, different size).     * If two objects are isomorphic:         * $V \propto L^3$         * $A \propto L^2$     * Example: If $V_1 = 1.0\,m^3$, $A_1 = 8.0\,m^2$, and $V_2 = 0.50\,m^3$, calculate $A_2$.         * $V_2/V_1 = (L_2/L_1)^3 \Rightarrow 0.5 = (L_2/L_1)^3 \Rightarrow L_2/L_1 = (0.5)^{1/3}$.         * $A_2 = A_1 \times (L_2/L_1)^2 = 8.0 \times (0.5)^{2/3} \approx 5.04\,m^2$.

Intensive vs. Extensive Quantities

• Context: Mass/Volume relationship.

• Formula: $\rho = \frac{m}{V}$.

• Scaling: If mass increases by a factor of 3 at constant density, volume also increases by a factor of 3 ($V_2 = 3V_1$).

• Extensive Quantities: Depend on the amount of matter (e.g., mass, volume).

• Intensive Quantities: Independent of the size or amount of matter (e.g., density).

Dimensional Analysis Example: Pendulum Period

• Observation: The period ($T$) of a simple pendulum depends only on the length of the string ($l$) and the acceleration due to gravity ($g$).

• Dimensional Goal: Find the relationship using dimensions.     * $[T] = [L]^a [LT^{-2}]^b$     * $[T^1] = [L^{a+b} T^{-2b}]$     * Equating exponents for $T$: $1 = -2b \Rightarrow b = -1/2$     * Equating exponents for $L$: $0 = a + b \Rightarrow a = -b = 1/2$     * Result: $T \propto \sqrt{\frac{l}{g}}$

Questions & Discussion

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• Question 3: Dimensional consistency check for various expressions of $s$ and $v$.

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• Example Problem (Ball on Incline): A ball rolls down an incline with constant acceleration from rest.     * Formula: $x = \frac{1}{2}at^2$.     * Given: It takes $5\,s$ to travel the length of 1 rod ($x_1$).     * Question: How long ($t_2$) to travel 3 rods ($x_2 = 3x_1$)?     * Proportionality: $x \propto t^2 \Rightarrow t \propto \sqrt{x}$.     * Solution: $t_2 = t_1 \sqrt{\frac{x_2}{x_1}} = 5\sqrt{3} \approx 8.66\,s$.