PHYS 2010 Zero-Assumption Companion: Structured Study Guide for College Physics (Non-Calculus)

Mathematics Review for Physics

Scientific Notation and Unit Conversion

Physics relies on concise mathematical language to express relationships among measurable quantities. Scientific notation is used to handle very large or small numbers efficiently, and unit conversion ensures answers are physically meaningful.

• Scientific Notation: Numbers are written as a value between 1 and 10 multiplied by a power of ten. Example: 4,500,000 = 4.5 × 106.

• Unit Conversion: Multiply by conversion factors arranged so unwanted units cancel. Area and volume conversions require squaring or cubing the conversion factor.

• SI Prefixes: Common prefixes include kilo (k, 103), centi (c, 10-2), milli (m, 10-3), micro (µ, 10-6), nano (n, 10-9).

• Significant Figures: Communicate measurement precision; round according to the least precise input.

Algebraic Rearrangement and Proportional Reasoning

Isolate the unknown symbol before substituting numbers. Proportional reasoning predicts how changes in one variable affect another.

• Direct Proportion: $y \propto x$ (y doubles if x doubles)

• Inverse Proportion: $y \propto 1/x$ (y halves if x doubles)

• Square Relation: $y \propto x^2$ (y quadruples if x doubles)

• Inverse Square: $y \propto 1/x^2$ (y becomes one fourth if x doubles)

Trigonometry and Vectors

Right-triangle trigonometry is essential for resolving vectors into components. Vectors have both magnitude and direction, while scalars have only magnitude.

• Component Formulas: $A_x = A \cos \theta$, $A_y = A \sin \theta$

• Pythagorean Theorem: $c^2 = a^2 + b^2$ for right triangles

Graphs and Physical Meaning

Graphs visually represent relationships. Slope and area have physical interpretations depending on the axes.

• Slope: $\Delta y / \Delta x$ (with units)

• Area under curve: Represents accumulated quantity (e.g., displacement, energy)

Models, Measurements, and Vectors

Modeling in Physics

Physics models simplify real situations by focusing on essential features and ignoring negligible details. Assumptions must be stated and checked for validity.

Measurement Quality

• Accuracy: Closeness to accepted value

• Precision: Repeatability of measurements

• Uncertainty: Range of doubt in a result

Vector Addition and Resultants

Vectors are added component-wise. The resultant vector's magnitude and direction are found using Pythagorean theorem and inverse tangent.

• Resultant Components: $R_x = \sum A_x$, $R_y = \sum A_y$

• Magnitude: $R = \sqrt{R_x^2 + R_y^2}$

• Direction: $\theta = \tan^{-1}(R_y / R_x)$

Motion Along a Straight Line (1D Kinematics)

Position, Velocity, and Acceleration

One-dimensional motion is described by position, velocity, and acceleration. Velocity is the rate of change of position; acceleration is the rate of change of velocity.

• Average Velocity: $v_{\text{avg}} = \frac{\Delta x}{\Delta t}$

• Average Acceleration: $a_{\text{avg}} = \frac{\Delta v}{\Delta t}$

Constant Acceleration Equations

For constant acceleration, use the kinematic equations:

• $v_f = v_i + a t$

• $\Delta x = v_i t + \frac{1}{2} a t^2$

• $v_f^2 = v_i^2 + 2 a \Delta x$

• $\Delta x = \frac{1}{2}(v_i + v_f)t$

Motion Graphs

Position, velocity, and acceleration graphs provide different perspectives on the same motion. Slope and area have specific meanings.

• x vs t: Slope = velocity

• v vs t: Slope = acceleration; area = displacement

• a vs t: Area = change in velocity

Motion in a Plane (2D Kinematics & Projectile Motion)

Projectile Motion

Projectile motion is described by independent horizontal and vertical components. Gravity affects only the vertical motion.

• Horizontal motion: $x = x_0 + v_{0x} t$

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

• Component velocities: $v_{0x} = v_0 \cos \theta$, $v_{0y} = v_0 \sin \theta$

Newton's Laws of Motion

Free-Body Diagrams and Force Analysis

Free-body diagrams (FBDs) are essential for analyzing forces acting on an object. Newton's laws connect force and motion.

• Newton's First Law: An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton's Second Law: $\sum F = m a$

• Newton's Third Law: For every action, there is an equal and opposite reaction (forces act on different objects).

Applications of Newton's Laws: Friction, Inclines, and Systems

Inclined Planes and Friction

On an incline, resolve weight into components parallel and perpendicular to the surface. Friction opposes motion.

• Weight components: $mg_{\text{parallel}} = mg \sin \theta$, $mg_{\text{perp}} = mg \cos \theta$

• Friction: $f_s \leq \mu_s N$, $f_k = \mu_k N$

Work, Energy, and Conservation

Work and Energy

Work is energy transferred by a force through displacement. Energy can change form but is conserved in ideal cases.

• Work: $W = F d \cos \theta$

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Potential Energy: $U_g = m g h$

• Conservation of Energy: $E_{\text{total}} = K + U$ (constant in ideal cases)

Momentum and Impulse

Momentum Conservation

Momentum is conserved in isolated systems. Impulse changes momentum.

• Momentum: $p = m v$

• Impulse: $J = F_{\text{avg}} \Delta t = \Delta p$

• Conservation: $\sum p_{\text{before}} = \sum p_{\text{after}}$

Rotational Motion and Dynamics

Linear vs Rotational Analogy

Rotational motion mirrors linear motion, but the distance from the axis matters. Angular quantities are used for rotation.

• Position vs Angle: $x \leftrightarrow \theta$

• Velocity vs Angular Velocity: $v \leftrightarrow \omega$

• Acceleration vs Angular Acceleration: $a \leftrightarrow \alpha$

• Force vs Torque: $F \leftrightarrow \tau$

• Momentum vs Angular Momentum: $p \leftrightarrow L$

• Kinetic Energy: $KE = \frac{1}{2} m v^2$, $KE_{\text{rot}} = \frac{1}{2} I \omega^2$

Torque and Rotational Inertia

• Torque: $\tau = r F \sin \phi$

• Rotational Inertia: $\sum \tau = I \alpha$

Elasticity and Periodic Motion

Hooke's Law and Simple Harmonic Motion

Springs obey Hooke's law in the elastic range. Simple harmonic motion (SHM) is periodic motion with restoring force proportional to displacement.

• Hooke's Law: $F_s = -k x$

• SHM Period: $T = 2 \pi \sqrt{\frac{m}{k}}$

Mechanical Waves and Sound

Wave Properties

Waves transport energy through a medium. Wavelength, frequency, and amplitude are key properties.

• Wave Speed: $v = f \lambda$

Fluid Mechanics

Pressure, Buoyancy, and Flow

Fluid mechanics covers density, pressure, hydrostatics, hydraulics, continuity, and Bernoulli's principle.

• Density: $\rho = \frac{m}{V}$

• Pressure: $P = \frac{F_{\text{perp}}}{A}$

• Hydrostatic Pressure: $P = P_{\text{surface}} + \rho g h$

• Buoyancy: $F_b = \rho_{\text{fluid}} g V_{\text{displaced}}$

• Continuity: $A_1 v_1 = A_2 v_2$

• Bernoulli: $P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$

Heat, Temperature, and Thermodynamics

Heat Transfer and Temperature

Heat is energy transferred due to temperature difference. Internal energy is stored microscopically.

• Specific Heat: $Q = m c \Delta T$

• Latent Heat: $Q = m L$

• Thermal Expansion: $\Delta L = \alpha L_0 \Delta T$

Thermal Properties of Matter and Ideal Gases

Ideal Gas Law and Kinetic Theory

• Ideal Gas Law: $P V = n R T$

• Kinetic Theory: $\langle K_{\text{trans}} \rangle = \frac{3}{2} k_B T$

• Internal Energy (monatomic): $U = \frac{3}{2} n R T$

The Second Law of Thermodynamics

Heat Engines, Efficiency, and Entropy

• First Law: $\Delta U = Q - W$

• Heat Engine Efficiency: $\eta = \frac{W}{Q_h} = 1 - \frac{Q_c}{Q_h}$

• Coefficient of Performance (COP): $COP_R = \frac{Q_c}{W}$, $COP_{HP} = \frac{Q_h}{W}$

• Entropy Change (reversible): $\Delta S = \frac{Q_{\text{rev}}}{T}$

• Carnot Efficiency Limit: $\eta_C = 1 - \frac{T_C}{T_H}$

Laboratory Preparation and Measurement

Measurement, Graphing, and Uncertainty

• Resolution: Smallest increment readable

• Accuracy: Closeness to reference

• Precision: Repeatability

• Uncertainty: Quantified doubt

• Graphing: Slope and intercept have physical meaning; units must be included

Quick Reference: Linear vs Rotational Quantities

Jonathan's Physics Problem-Solving Chain

Structured Approach to Physics Problems

Follow a systematic chain for solving physics problems:

• Read the problem

• Draw the situation

• List knowns and find unknowns

• Choose the governing principle

• Select the equation based on the principle

• Calculate with units

• Check and explain the result

Choose the physics principle before choosing the formula.