AP Physics C: Mechanics — Cheatsheet

Formulas & relationships

On the exam

• On the AP exam, given x(t) you should be able to produce v(t) and a(t) instantly by differentiating, and read the reverse too: where the x–t slope is zero the object is momentarily at rest, and where the v–t slope is zero the acceleration is zero.
• Keep the subscript bookkeeping strict: write every velocity as v(object/frame). The chain v(A/C) = v(A/B) + v(B/C) only works when the inner labels match, and swapping any pair introduces a sign flip. Getting the subscripts right turns most relative-motion questions into simple addition.
• Read the question for **displacement** versus **distance** before computing anything. If any part of the velocity graph lies below the axis the two answers differ, and the difference is usually the entire point of the item.
• On the AP exam, draw the free-body diagram first and tilt your axes to match the motion. Write ΣF = ma along each axis separately. Getting the components of weight right — mg sin θ along a slope, mg cos θ into it — is where most points are won or lost.
• Keep the two formulas straight: F = Gm₁m₂/r² is the force between two masses, while g = GM/R² is the field (acceleration) one mass produces at distance R. Both are inverse-square — whenever a distance changes, square the ratio before scaling.
• N = mg only on a horizontal surface with no vertical component of applied force. On an incline N = mg cos θ; with a push at an angle, N changes; in an accelerating elevator, N changes. Deriving N from the perpendicular equilibrium condition every time costs ten seconds and prevents a whole family of errors.
• On any apparent-weight question, write ΣF = ma with a stated sign convention before substituting numbers. Deciding "heavier or lighter" by intuition works about as often as it fails; the equation N = m(g + a) with a signed correctly never does.
• Move fluently in both directions: integrate a conservative force to get U (U = −∫F dx), and differentiate U to get the force (F = −dU/dx). On a U-versus-x graph, remember the force points downhill and equilibria sit where the slope is zero.
• Use P = W/Δt when you know a total amount of work over an interval, and P = Fv (or F·v) when you want the power at a specific instant or speed. A force perpendicular to the velocity — like the centripetal force on a satellite — does zero work and delivers zero power.
• A force perpendicular to displacement does zero work. That single fact disposes of the normal force on a level surface, the tension in a conical pendulum, the magnetic force on a charge, and the centripetal force in any circular orbit — all common exam targets.
• Free-response energy questions are graded on the **accounting**, not the arithmetic. Write an explicit initial-equals-final statement naming every term before substituting numbers. A correct number arrived at without a stated energy equation routinely loses the setup points; a stated equation with a small numerical slip usually keeps most of them.
• For a continuous body, the recipe is always the same: write dm using the density, integrate x dm for the numerator and dm for the total mass M, then divide. Never average the endpoints — that only works for a uniform object.
• The rocket equation comes from momentum conservation with changing mass, not from F = ma with constant mass. Watch the logarithm: the payoff for carrying more fuel diminishes, since Δv grows only as ln(m_i/m_f).
• Set up a small table with columns for x and y and rows for before and after. Filling it in before doing any algebra prevents the dominant error on these problems, which is not conceptual but bookkeeping — a dropped component or a sign flipped on one object.
• Two objects with identical mass can have very different moments of inertia. Always ask where the mass sits relative to the axis: the r² weighting in I = ∫r² dm rewards mass at the rim and penalizes it near the center.
• For rolling problems, always bring in the constraint a_cm = Rα to connect the force equation to the torque equation. Remember that the shape factor I/MR² alone decides the race down an incline — mass and radius drop out.
• Free-response equilibrium problems award points for the **setup**: a labeled free-body diagram, an explicit statement of the pivot chosen, and the torque equation with moment arms identified. Students who go straight to numbers routinely lose those points even when the final answer is right.
• Keep L = Iω and τ = dL/dt distinct: one is the angular momentum, the other is the rule for how torque changes it. The single most useful consequence is that zero net external torque forces angular momentum to stay constant.
• For circular orbits set gravity equal to the centripetal force (GMm/r² = mv²/r). For elliptical orbits, use conservation of angular momentum (v_p r_p = v_a r_a) at the apses and conservation of energy between them — energy and angular momentum together pin down the whole orbit.
• When a rolling problem appears, immediately identify k = I/MR² and reach for a = g sin θ/(1 + k). Deriving it from scratch each time is a valid but slow route, and the derivation is worth doing once carefully so the formula can be used with confidence thereafter.
• State the small-angle approximation explicitly on free-response: sin θ ≈ θ is what turns d²θ/dt² = −(g/L)sin θ into SHM. Remember the simple-pendulum period is independent of mass, and use the physical-pendulum formula T = 2π√(I/mgd) whenever the swinging object is an extended body.
• Resonance is defined by the **driving** frequency matching the natural frequency, not by any property of the driving force's strength. A weak force applied at resonance can produce a larger response than a strong force applied far from it — which is the whole reason resonance is worth a name.

How to get a 5

• Always define your coordinate system and direction of positive rotation. Consistency between linear (a) and rotational (α) signs is crucial in rolling problems.
• Don't memorize specific derived formulas; practice deriving them from Newton's Laws and Energy Conservation. You will need to show the steps on the FRQ.
• In variable mass or density problems, always relate dm to dx (or dr) using density (λ, σ, ρ). e.g., dm = λ dx.
• Check your boundary conditions and limits of integration carefully when doing work or center of mass integrals.
• Check first whether a quantity is constant. If a, F, or λ depends on t, x, or r, then the constant-acceleration formulas and W = Fd are off the table — write the derivative or the integral instead, and say explicitly what variable you are integrating over.
• Show every integral with its limits and its dm or dt element before evaluating. Rubrics award points for a correct integrand and correct limits even when the arithmetic then goes wrong, while a bare numerical answer earns nothing.
• Turn “derive” prompts into named steps: state the principle (Newton’s second law, conservation of angular momentum), write it symbolically for this system, apply the constraint (a = αR, v = ωR, inextensible string), then solve for the requested symbol, leaving the answer in the given variables.
• Check limits and units on every symbolic result. For a = mg/(m + M/2), confirm a → g as M → 0 and a → 0 as M → ∞. A limit that misbehaves reveals an algebra error faster than re-reading each line.
• Graph-sketching questions want features, not artistry: correct intercept, correct initial slope, correct curvature, and a labeled asymptote. For exponential approaches, mark the value at one time constant (63% of the total change) and show the curve flattening rather than touching the asymptote.

Key terms