Physics Comprehensive Checklist & Flashcards

Core Principle: Your final answer cannot look more precise than the least precise measurements given. Non-zero numbers, zeros sandwiched between non-zero digits, and trailing zeros behind decimal placements are significant.

Derivation Strategy: To go from kilometers per hour to meters per second, multiply by 1,000 meters and divide by 3,600 seconds. This simplifies directly down into dividing your value by 3.6.

Warning: Never round off values halfway through multi-stage calculations. Keep unrounded figures saved in your calculator memory and round to the correct sig figs only at the final step to prevent calculation drift.

Problem: Convert 110 km/h using 3 sig figs.
Solution: 110 / 3.6 = 30.555... m/s.
Rounding to 3 significant figures yields 30.6 m s^-1.

Core Principle: Orthogonal vector fields (x and y axis systems) act completely independently of one another. You must resolve, sum, and manipulate x components and y components separately before combining them.

Formulas: For a vector R at angle θ from the x-axis:
• R_x = R * cos(θ)
• R_y = R * sin(θ)
• Total Mag = √( (ΣR_x)² + (ΣR_y)² )

Warning: Never add the raw magnitudes of two angled vectors directly together (e.g., 5N + 5N does not equal 10N unless they point in the exact same direction). Always use vector component tracking.

Scenario: Force 1 is 3.0 N East (Σx=3, Σy=0). Force 2 is 4.0 N North (Σx=0, Σy=4).
Result: Net Magnitude = √(3² + 4²) = √25 = 5.0 N.

Core Principle: An FBD maps real, external forces physically acting on an object. Weight pulls toward Earth's center; Normal force pushes square away from the ramp plane; Friction runs parallel to the surface.

Orientation Geometry:
• Weight (W): Points straight down vertical.
• Normal Force (F_N): Angle matching tilt, points 90° away from the slope.
• Friction (F_f): Tracks flush along the surface line.

Warning: Never draw the Normal force pointing straight up on an incline. "Normal" means mathematically perpendicular to the plane of touch. If the surface tilts, your Normal force vector must tilt.

Scenario: Box sliding down a ramp.
Your Diagram Checklist: Draw a box. Draw an arrow straight down for W. Draw an arrow 90° from the ramp up for F_N. Draw an arrow pointing back up the ramp surface for F_f.

Core Principle: Tilting your axes parallel (||) and perpendicular (⊥) to the ramp surface makes math easier because motion happens purely along the parallel plane, while the perpendicular forces net to zero.

Formulas: Using plane angle θ:
• F_g,|| = m * g * sin(θ) [pulls down ramp]
• F_g,⊥ = m * g * cos(θ) [presses into ramp]

Warning: On standard flat vectors, horizontal uses cosine. On an inclined plane, the component parallel to the surface uses sine. Do not mix them up during step conversions.

Problem: 5.0 kg mass on a 30° slope.
• F_g,|| = 5 * 9.8 * sin(30°) = 24.5 N.
• F_g,⊥ = 5 * 9.8 * cos(30°) = 42.4 N.

Core Principle: Friction force equals friction coefficient μ times the Normal Force F_N. Because the Normal force drops as a ramp tilts steeper, the friction force changes depending on the angle.

Ramp Balancing Formulas:
• F_N = m * g * cos(θ)
• F_f = μ * m * g * cos(θ)
• F_net,|| = m * g * sin(θ) - F_f = m * a

Warning: F_N ONLY equals mg on flat horizontal surfaces. If an object sits on an inclined plane, using F_N = mg will overcalculate friction forces and lead to wrong answers.

Problem: Object slides down ramp with θ=30°, μ=0.20.
a = g*sin(30°) - μ*g*cos(30°)
a = 4.90 - 1.70 = 3.20 m s^-2.

Core Principle: Acceleration due to gravity (9.80 m/s²) means that, in a vacuum, a falling object gains exactly 9.80 m/s of downward speed for every elapsed second of flight.

Formula Behavior: Because a = Δv/Δt is constant under gravity, velocity shifts linearly (v = u + at), but the distance traveled per second grows quadratically (s = ut + ½at²).

Warning: Do not confuse speed with acceleration. A falling object's speed increases continuously, but its acceleration remains a constant 9.80 m/s² downwards.

Scenario: Object dropped from rest.
• Velocity after 1s = 9.8 m/s.
• Velocity after 2s = 19.6 m/s.
• Total acceleration at all points = 9.8 m/s².

Core Principle: On a velocity-time graph, the slope (rise over run) represents acceleration. A flat line means constant velocity (zero acceleration); a straight sloped line means uniform acceleration.

Formula: For a segment bounded by coordinates (t₁, v₁) and (t₂, v₂):
a = (v₂ - v₁) / (t₂ - t₁)

Warning: Always check if a graph plots speed vs time or distance vs time. Finding the slope of a distance-time graph yields velocity, not acceleration. Check your axes labels carefully.

Scenario: Speed climbs from 0 to 20 m/s over 4 seconds.
Solution: Acceleration gradient = (20 - 0) / (4 - 0) = 5.0 m s^-2.

Core Principle: Projectiles experience zero horizontal acceleration (speed stays constant), while vertical motion is subject to constant gravitational drop. Time connects both component tracks.

Formulas:
• Horizontal: s_x = v_x * t
• Vertical: s_y = u_y*t + ½*a_y*t²
For horizontal launches, u_y = 0.

Warning: Never substitute horizontal velocity parameters into vertical acceleration equations. Keep your variables sorted in separate tables of data.

Problem: Dart thrown horizontally at 15 m/s, hits ground in 0.40s.
Solution: Horizontal Distance = v_x * t = 15 * 0.40 = 6.0 m.

Core Principle: Every gravitational pull is an interactive pairing. The force exerted by a large planet on a tiny moon is exactly equal in magnitude to the force exerted by the tiny moon back on the large planet.

Formula Behavior: F = G*M*m / r². The force is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centers.

Warning: Equal forces do not mean equal accelerations. Because a = F/m, the less massive object will accelerate much more noticebly than the massive object, despite experiencing the same force.

Scenario: Planet exerts 500 N force on a satellite.
Question: How much force does the satellite exert on the planet?
Answer: Exactly 500 N.

Core Principle: When a force changes over a distance, you cannot use Work = F * s. You must find the geometric area underneath the Force-Displacement graph to obtain total work.

Formulas:
• Triangular Area = ½ * base * height
• Work Total = ΔK_e = K_f - K_i
• Kinetic Energy = ½ * m * v²

Warning: For a triangular force graph ramping up to 20N over 5m, do not multiply 20N * 5m = 100J. A triangle takes up half that area, so it must be ½ * 20 * 5 = 50J.

Problem: Graph area equals 16 J of work done on a 2.0 kg particle starting from rest. Find v.
Solution: 16 = ½ * (2.0) * v² → v² = 16 → v = 4.0 m s^-1.

Core Principle: Potential energy is work stored due to position. While Joules (J) is the standard SI unit, subatomic problems use electron-volts (eV). 1 eV = 1.60 × 10^-19 J.

Formula: ΔU_g = m * g * Δh = m * g * (h_final - h_initial). Potential energy changes depend purely on the net vertical displacement, not the path taken.

Warning: If an object falls or moves downward, its change in potential energy is negative (ΔU is negative). Always subtract initial height from final height.

Problem: 2.0 kg block drops from h=10m down to h=3m.
Solution: ΔU = 2.0 * 9.8 * (3 - 10) = 2.0 * 9.8 * (-7) = -137.2 J.

Core Principle: In any isolated system with no external forces, total initial momentum equals total final momentum (Σp_i = Σp_f). This holds true independently for x and y components.

Formulas: p = m * v.
• Component split:
Σm*v_x,initial = Σm*v_x,final
Σm*v_y,initial = Σm*v_y,final

Warning: Momentum is a vector. If Ball A moves right (+v) and Ball B moves left (-v), you must include the negative sign in your momentum sum.

Problem: 2kg block moving at 6m/s hits and sticks to a stationary 4kg block.
Solution: (2*6) + 0 = (2+4)*v_f → 12 = 6*v_f → v_f = 2.0 m s^-1.

Core Principle: Adding thermal energy either speeds up molecules (rising temperature) OR breaks intermolecular bonds (phase change at a constant temperature plateau). They never happen at the same time.

Formulas:
• Temperature change: Q = m * c * ΔT
• Phase transformation: Q = m * L
(where L is latent heat of fusion or vaporization).

Warning: To find the total heat needed to change -10°C ice into 20°C water, you cannot do a single calculation. You must sum three separate steps: heating ice, melting ice, and heating liquid water.

Scenario: A heating graph shows a substance's temperature flattening out out at 65°C before rising again.
Analysis: The 65°C flat plateau marks the exact melting point of that substance.

Core Principle: For an ideal gas, macroscopic properties (Pressure, Volume, Temperature) are related to the total count of particles (moles, n) via the universal gas constant R.

Formula: P * V = n * R * T.
• R = 8.314 J mol^-1 K^-1.
Crucial step: Temperature T MUST always be converted to Kelvin (K = °C + 273.15).

Warning: Never use temperature in Celsius inside PV = nRT. For example, dividing by 0°C creates an invalid mathematical operation, whereas converting it to 273K keeps the math correct.

Problem: Find pressure for 2.0 moles at 300K in a 0.05 m³ tank.
Solution: P = nRT/V = (2 * 8.314 * 300) / 0.05 = 49884 / 0.05 = 9.98 × 10⁵ Pa.

Core Principle: Wavelength is the length of one full repeating wave cycle. Amplitude is the maximum displacement measurement, representing the total wave energy.

Measurements:
• λ = horizontal distance from one crest to the next consecutive crest.
• A = vertical height from the center equilibrium line up to a crest.

Warning: Do not measure amplitude from the bottom trough to the top crest. That is peak-to-peak height. Amplitude is exactly half of that total vertical distance.

Scenario: A wave profile graph shows a total vertical distance of 6.0 cm from top to bottom, and 10 cm horizontally between crest 1 and crest 3.
Values: Amplitude = 3.0 cm; Wavelength = 10/2 = 5.0 cm.

Core Principle: Mechanical waves pass energy via particle collisions, meaning they cannot travel through empty space. Electromagnetic waves consist of oscillating fields and require no physical medium.

Constants: All EM waves move through a vacuum at the speed of light: c = 3.00 × 10⁸ m/s. Mechanical wave speeds depend entirely on the density and elasticity of the material they travel through.

Warning: Do not assume radio waves or microwaves are mechanical because we can't see them. They belong to the electromagnetic spectrum and can travel through outer space. Sound waves, however, are mechanical.

Question: Sort ultrasound, gamma rays, and sound waves.
• Mechanical: Sound, Ultrasound.
• Electromagnetic: Gamma Rays.

Core Principle: Real images form where light rays physically intersect, allowing them to be projected onto a screen. Virtual images form where light rays appear to diverge from behind a lens or mirror.

Formulas:
• 1/f = 1/d_o + 1/d_i
• M = h_i / h_o = -d_i / d_o
(Negative d_i indicates a virtual image).

Warning: For concave/diverging lenses and convex mirrors, the focal length f is always negative. Forgetting this sign will flip your calculations and lead to incorrect answers.

Problem: Object at d_o=6cm, focal length f=2cm. Find d_i.
Solution: 1/d_i = 1/2 - 1/6 = 3/6 - 1/6 = 2/6 = 1/3.
Inverting gives d_i = 3.0 cm (Real image).

Core Principle: Light bends toward the normal line when entering a optically denser medium (higher refractive index, n). Total Internal Reflection occurs only when light travels from a high-n medium toward a low-n medium and exceeds the critical angle.

Formulas:
• Snell's Law: n₁ * sin(θ₁) = n₂ * sin(θ₂)
• Critical Angle: sin(θ_c) = n₂ / n₁

Warning: Always ensure your calculator is set to Degree mode (DEG) rather than Radian mode (RAD) before computing trigonometric values for optics problems.

Problem: Find θ_c from glass (n=1.50) into air (n=1.00).
Solution: sin(θ_c) = 1.00 / 1.50 = 0.6667.
θ_c = sin^-1(0.6667) = 41.8°.

Core Principle: Objects become charged via electron transfer (friction or contact). Like charges repel each other, while opposite charges experience an attractive force.

Formula: F = k * |q₁ * q₂| / r².
• Coulomb constant k = 9.00 × 10⁹ N m² C^-2.
Doubling the separation distance drops the force to one-quarter of its initial value.

Warning: Pay close attention to unit prefixes like microcoulombs (μC) or nanocoulombs (nC). Convert them to standard Coulombs (1 μC = 10^-6 C) before plugging them into the formula.

Problem: Two +1.0 C charges separated by 3.0 meters.
Solution: F = (9.00 × 10⁹ * 1.0 * 1.0) / 3.0² = 9.00 × 10⁹ / 9.0 = 1.00 × 10⁹ N.

Core Principle: Series resistors share the exact same current pathway. Parallel resistor networks branch out, sharing the same voltage drop across their branches.

Formulas:
• Series: R_eq = R₁ + R₂ + ...
• Parallel: 1/R_eq = 1/R₁ + 1/R₂ + ...
Ohm's Law: V = I * R.

Warning: When computing parallel tracks, don't stop after adding fractions. If 1/R_eq = 1/4, you must invert that value to get your final answer: R_eq = 4 Ω.

Problem: Combine 6.0 Ω and 3.0 Ω in parallel.
Solution: 1/R = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2.
Inverting gives R_eq = 2.0 Ω.

Core Principle: Coiling a current-carrying wire wraps individual magnetic loops together. This builds a strong, uniform magnetic field inside the core of the solenoid that acts like a bar magnet.

Formula: B = μ₀ * N * I / L.
• μ₀ = 4π × 10^-7 T m A^-1.
• N = total turn count; L = core length in meters.

Warning: Pay close attention to whether the problem gives you the total turn count (N) or the turn density per meter (n). If turn density is given, n = N/L, so B = μ₀*n*I.

Problem: Solenoid has 500 turns, length 0.20m, current 4.0A.
Solution: B = (4π×10^-7 * 500 * 4.0) / 0.20 = 0.00251 / 0.20 = 1.26 × 10^-2 T.

Core Principle: A magnetic force always acts perpendicular to a moving particle's velocity vector. This constant sideways deflection acts as a centripetal force, forcing the charged particle into a circular orbital path.

Formulas: For orthogonal paths:
• F_mag = q * v * B
• Equating forces: q * v * B = m * v² / r
• Simplifying for radius: r = m * v / (q * B)

Warning: When using the Right-Hand Rule to find the force direction, remember it applies directly to positive charges. For negative charges (like electrons), you must reverse the final force arrow direction.

Problem: Electron (m=9.11×10^-31kg, q=1.6×10^-19C) loops at v=2×10⁶m/s with r=0.1m. Find B.
Solution: B = mv/qr = 1.82×10^-24 / 1.6×10^-20 = 1.14 × 10^-4 T.

Core Principle: Atoms of an element have the same number of protons but can vary in their neutron count (isotopes). Standard nuclide notation explicitly identifies these subatomic counts.

Formatting Structure: ^A_ZX
• Z = Atomic number (protons).
• A = Mass number (protons + neutrons).
• Neutron count N = A - Z.

Warning: The top mass number (A) is not the neutron count. It represents the total sum of protons and neutrons combined. You must subtract Z from A to isolate the number of neutrons.

Example element: ¹⁴₆C.
• Proton count Z = 6.
• Total nucleons A = 14.
• Neutron count N = 14 - 6 = 8 neutrons.

Core Principle: A half-life (t_½) is the fixed time required for exactly half of the remaining unstable radioactive parent nuclei in a sample to decay away.

Formulas:
• Remaining Amount: N(t) = N₀ * (0.5)ⁿ
• Half-life steps: n = t / t_½
This model tracks remaining mass, activity rates, or overall particle counts.

Warning: Radioactive decay is exponential, not linear. If a sample loses half its mass in 5 days, it will not lose the remaining half in the next 5 days. It will lose half of what is left (dropping to one-quarter total).

Problem: 80g sample has t_½ = 3 hours. How much is left after 9 hours?
Solution: Steps n = 9 / 3 = 3 half-lives.
Mass = 80 * (0.5)³ = 80 / 8 = 10 grams.

Core Principle: During nuclear transformations, total mass number (ΣA) and total atomic number (ΣZ) must balance perfectly across both sides of the reaction equation.

Algebra Setup: For ^A1_Z1J + ^A2_Z2K → ^A3_Z3L + ^A4_Z4X:
• Top balance: A1 + A2 = A3 + A4
• Bottom balance: Z1 + Z2 = Z3 + Z4

Warning: Gamma rays (y) carry no mass or charge (⁰₀y). They are high-energy photons emitted to release excess energy, so they do not alter the nucleon count balances during your calculations.

Problem: Balance ²₁H + ³₁H → ⁴₂He + ^A_ZX.
• Top: 2 + 3 = 4 + A → A = 1.
• Bottom: 1 + 1 = 2 + Z → Z = 0.
¹₀X matches a neutron (n).

Core Principle: Light hits a metal surface as quantized energy packets called photons. If a photon's frequency is below the threshold frequency (work function), no electrons will be ejected, regardless of the light's intensity.

Formulas: E_photon = h * f.
• Einstein Equation:
K_max = h*f - h*f₀ = ½ * m_e * v²
(where h = 6.63×10^-34 J s, and hf₀ is the work function).

Warning: Increasing the intensity (brightness) of light only increases the number of emitted electrons, not their kinetic energy. Only increasing the light's frequency increases electron speed.

Problem: hf incoming = 5.0×10^-19J. Work function threshold energy is 3.5×10^-19J.
Solution: K_max = 5.0×10^-19 - 3.5×10^-19 = 1.5 × 10^-19 J.