PHYS 206 Exam 1 Review: Chapters 1–5 Study Guide

Units, Physical Quantities, and Vectors

Units and Significant Figures

Physical quantities are measured using standardized units, and calculations must maintain unit consistency and proper precision.

• SI Units: Length (meter, m), Time (second, s), Mass (kilogram, kg).

• Prefixes: Used to scale units by powers of ten. See table below.

• Unit Consistency: Only quantities with the same units can be added, subtracted, or equated.

• Unit Conversion: Use conversion factors (e.g., $\frac{60\ s}{1\ min}$) and raise the entire factor to a power when converting squared or cubed units.

• Significant Figures: For multiplication/division, keep as many significant figures as the least precise factor; for addition/subtraction, round to the least precise decimal place. Round only the final result.

Prefix	Symbol	Factor
nano	n	$10^{-9}$
micro	μ	$10^{-6}$
milli	m	$10^{-3}$
centi	c	$10^{-2}$
kilo	k	$10^{3}$
mega	M	$10^{6}$
giga	G	$10^{9}$

Scalars and Vectors

Scalars are single numbers with units; vectors have both magnitude and direction.

• Scalars: Examples include mass, time, distance, speed.

• Vectors: Examples include displacement, velocity, acceleration, force.

• Vector Operations: Vector ± vector and scalar × vector yield vectors; dot and cross products combine two vectors.

• Dot Product: $\vec{A} \cdot \vec{B}$ yields a scalar.

• Cross Product: $\vec{A} \times \vec{B}$ yields a vector.

Components, Direction Angles, and Unit Vectors

Vectors are described by their components along chosen axes and unit vectors.

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

• Unit Vectors: $\hat{i}$, $\hat{j}$, $\hat{k}$ point along +x, +y, +z axes.

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

• Direction Angles: $A_x = A \cos \theta$, $A_y = A \sin \theta$ (for xy-plane).

• Compass Directions: "30° east of north" means start from north and rotate 30° toward east.

Adding and Subtracting Vectors

Vectors add graphically head-to-tail and algebraically by components.

• Component Addition: $C_x = \alpha A_x + \beta B_x$

• Magnitude Bounds: $|A - B| \leq |A + B| \leq A + B$

• Magnitude of Sum: $|\vec{A} \pm \vec{B}|^2 = A^2 + B^2 \pm 2AB \cos \theta$

The Dot (Scalar) Product

The dot product measures how much two vectors point in the same direction.

• Formula: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z = AB \cos \theta$

• Applications: Angle between vectors, test for perpendicularity ($\vec{A} \cdot \vec{B} = 0$), work ($W = \vec{F} \cdot \vec{d}$).

The Cross (Vector) Product

The cross product yields a vector perpendicular to both original vectors.

• Magnitude: $|\vec{A} \times \vec{B}| = AB \sin \theta$

• Direction: Right-hand rule.

• Component Formula: $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y) \hat{i} + (A_z B_x - A_x B_z) \hat{j} + (A_x B_y - A_y B_x) \hat{k}$

• Properties: Reversing order reverses sign; vanishes for parallel vectors.

• Scalar Triple Product: $((\vec{A} \times \vec{B}) \cdot \vec{C})$ gives volume of parallelepiped.

Motion Along a Straight Line

Position, Velocity, and Acceleration

Motion along a line is described by position, velocity, and acceleration as functions of time.

• Position: $x(t)$

• Displacement: $\Delta x = x_2 - x_1$

• Average Velocity: $\bar{v}_x = \frac{\Delta x}{\Delta t}$

• Average Speed: $\text{distance traveled} / \Delta t$

• Instantaneous Velocity: $v_x = \frac{dx}{dt}$

• Instantaneous Acceleration: $a_x = \frac{dv_x}{dt} = \frac{d^2x}{dt^2}$

Calculus and Initial Conditions

Integration relates acceleration, velocity, and position, with constants determined by initial conditions.

• Velocity from Acceleration: $v_x(t) = v_{0x} + \int_0^t a_x(t') dt'$

• Position from Velocity: $x(t) = x_0 + \int_0^t v_x(t') dt'$

• Average Velocity: $\bar{v}_x = \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} v_x dt$

Constant Acceleration and Free Fall

For constant acceleration, motion equations simplify and describe free fall near Earth's surface.

• Equations:

  • $v_x = v_{0x} + a_x t$

  • $x - x_0 = v_{0x} t + \frac{1}{2} a_x t^2$

  • $v_x^2 = v_{0x}^2 + 2 a_x (x - x_0)$

  • $x - x_0 = \frac{1}{2} (v_{0x} + v_x) t$

• Free Fall: $a_y = -g$ (with $g \approx 9.8\ m/s^2$)

• Time to Top: $t = v_0 / g$

• Maximum Height: $h = \frac{v_0^2}{2g}$

Motion in Stages and Two Objects

When acceleration changes, apply constant-acceleration equations to each stage. For two objects, use one axis and clock.

• Relative Velocity: $v_{rel} = v_{2x} - v_{1x}$

• Relative Acceleration: $a_{rel} = a_{2x} - a_{1x}$

Graphs of Motion and Motion Diagrams

Graphical representations help visualize motion.

• x–t Graph: Height = position; slope = velocity.

• v_x–t Graph: Slope = acceleration; area under curve = displacement.

• a_x–t Graph: Area = change in velocity.

• Motion Diagrams: Spacing between positions indicates speeding up or slowing down.

Motion in Two or Three Dimensions

Position, Velocity, and Acceleration Vectors

Motion in space is described by vector quantities.

• Position: $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$

• Velocity: $\vec{v} = \frac{d\vec{r}}{dt}$

• Acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$

• Speed Change: Parallel component of $\vec{a}$ changes speed; perpendicular component changes direction.

Projectile Motion

Projectile motion combines independent horizontal and vertical motions under gravity.

• Horizontal Acceleration: $a_x = 0$

• Vertical Acceleration: $a_y = -g$

• Equations:

  • $x = x_0 + v_{0x} t$

  • $v_x = v_{0x}$

  • $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

  • $v_y = v_{0y} - g t$

  • $v_y^2 = v_{0y}^2 - 2g(y - y_0)$

• Time of Flight (equal heights): $t = \frac{2 v_0 \sin \theta}{g}$

• Range: $R = \frac{v_0^2 \sin 2\theta}{g}$

• Path: Parabolic.

Relative Velocity

Relative velocity describes how motion appears from different reference frames.

• General Formula: $\vec{v}_{P/A} = \vec{v}_{P/B} + \vec{v}_{B/A}$

• Applications: Boat in current, airplane in wind, launch from moving vehicle.

Circular Motion

Circular motion involves constant turning of velocity, requiring inward acceleration.

• Uniform Circular Motion:

  • Radial acceleration: $a_{rad} = \frac{v^2}{R} = \frac{4\pi^2 R}{T^2}$

  • Speed: $v = \frac{2\pi R}{T}$

• Nonuniform Circular Motion:

  • Tangential acceleration: $a_{tan} = \frac{d|\vec{v}|}{dt}$

  • Total acceleration: $|\vec{a}| = \sqrt{a_{rad}^2 + a_{tan}^2}$

  • Angle: $\tan \phi = \frac{|a_{tan}|}{a_{rad}}$

• Released Object: Leaves along tangent with instantaneous velocity.

Newton’s Laws of Motion

Force and Newton’s First Law

Force is a vector quantity; Newton's first law defines inertial frames.

• Force: Push or pull; measured in newtons ($1\ N = 1\ kg \cdot m/s^2$).

• Net Force: $\vec{F}_{net} = \sum \vec{F}$

• First Law: $\sum \vec{F} = \vec{0} \iff \vec{a} = \vec{0}$

• Inertial Frame: Frame where Newton's laws hold.

Newton’s Second Law

Acceleration is proportional to net force and inversely proportional to mass.

• Formula: $\sum \vec{F} = m \vec{a}$

• Component Form: $\sum F_x = m a_x$, $\sum F_y = m a_y$, $\sum F_z = m a_z$

• Mass: Positive scalar measuring inertia.

Newton’s Third Law

For every action, there is an equal and opposite reaction.

• Law: $\vec{F}_{AB} = -\vec{F}_{BA}$

• Third-Law Pair: Forces act on different objects, are equal in magnitude and opposite in direction.

• Examples: Rocket and exhaust, walker and ground.

Weight, Normal Force, and Apparent Weight

Weight is the gravitational force; normal force is the perpendicular push of a surface.

• Weight: $\vec{F}_G = m \vec{g}$

• Normal Force: $\vec{F}_N$; determined by Newton's second law perpendicular to surface.

• Apparent Weight: What a scale reads; $F_N = m(g + a_y)$ in an elevator.

Free-Body Diagrams

Free-body diagrams show all forces acting on an object.

• Draw each force, label by type and agent.

• Choose axes, resolve forces into components.

• Apply Newton's second law along each axis.

• Add constraints and friction relations.

Tension and Connected Objects

Tension is the pull along a rope; ideal ropes and pulleys simplify analysis.

• Tension: $F_T$; same throughout ideal rope.

• Connected Objects: Share acceleration; write Newton's second law for each.

• Examples: Atwood machine, movable pulley, blocks in contact.

Inclines and Equilibrium

On an incline, resolve forces along and perpendicular to the slope.

• Gravity Components: $mg \sin \theta$ down slope, $mg \cos \theta$ into slope.

• Normal Force: $F_N = mg \cos \theta$

• Acceleration: $a = g \sin \theta$ (frictionless)

• Equilibrium: $\sum F_x = \sum F_y = \sum F_z = 0$

Applying Newton’s Laws

Static and Kinetic Friction

Friction opposes sliding or the tendency to slide; depends on normal force.

• Kinetic Friction: $F_k = \mu_k F_N$

• Static Friction: $F_s \leq \mu_s F_N$

• Direction: Opposes motion or tendency to move.

• Transition: Static friction grows with applied force up to maximum; then kinetic friction applies.

Friction on an Incline

Friction acts along the slope, opposing sliding or tendency to slide.

• Normal Force: $F_N = mg \cos \theta$

• At Rest: $F_s = mg \sin \theta$ (must be $\leq \mu_s mg \cos \theta$)

• Sliding: Kinetic friction balances downhill component at constant velocity.

Connected Objects with Friction

Friction is considered separately for each object; analyze using Newton's second law.

• Block on Table and Hanging Block: Movement starts if hanging weight exceeds maximum static friction.

• Stacked Blocks: Each block's free-body diagram includes all surfaces it touches; friction limits acceleration.

Dynamics of Circular Motion

Circular motion requires an inward (radial) net force; tangential force changes speed.

• Radial Force: $\sum F_{rad} = m \frac{v^2}{R}$

• Tangential Force: $\sum F_{tan} = m \frac{dv}{dt}$

• Flat Curve: Static friction supplies inward force; maximum static friction must be at least $m v^2 / R$.

• Banked Curve: Normal force tilts toward center; at design speed, horizontal component supplies inward force.

• Conical Pendulum: Vertical component of tension balances weight; horizontal component supplies inward force.

• Vertical Circle: At top, gravity and tension point toward center; minimum speed required for tension to be nonzero.

• Apparent Weight: Over a hill, normal force is less than $mg$; at high speed, object may lose contact.

Appendix: Optional Extensions

Volume from a Scalar Triple Product

The volume of a parallelepiped spanned by three vectors is given by the absolute value of the scalar triple product.

• Formula: $V = |(\vec{A} \times \vec{B}) \cdot \vec{C}|$

Radius of Curvature of a General Path

Any curved path can be locally approximated by a circle; the radius of curvature relates to perpendicular acceleration.

• Formula: $a_{\perp} = \frac{v^2}{R}$, $a_{\parallel} = \frac{dv}{dt}$

• Radius: $R = \frac{v^2}{a_{\perp}}$

Drag and Terminal Speed

Drag opposes motion through a fluid; terminal speed is reached when drag balances weight.

• Linear Drag: $v_T = \frac{mg}{b}$

• Quadratic Drag: $v_T = \sqrt{\frac{mg}{D}}$