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Formula for work done by a constant force?
Answer: W = F·d·cos(θ)
Units: Joules (J = N·m = kg·m²/s²). θ is the angle between the force and displacement vectors. Trap: if a force is perpendicular to displacement (normal force, centripetal force), it does ZERO work even though the force is nonzero.
Work-Energy Theorem?
Answer: W_net = ΔKE = KE_f − KE_i
Units: Joules. Trap: you must use the NET work (sum of work done by all forces), not the work of a single force, unless that force is the only one acting.
How do you find work from a Force vs. position graph, and what is the work to stretch/compress a spring?
Answer: W = area under the F–x graph; for an ideal spring, W = ½kx²
Units: Joules. x is measured from the spring's natural (equilibrium) length. Trap: don't confuse x (displacement from equilibrium) with the spring's total length.
Formula for kinetic energy?
Answer: KE = ½mv²
Units: Joules (kg·m²/s²). Trap: KE depends on v², so doubling speed quadruples KE, not doubles it.
Formula for gravitational potential energy?
Answer: U_g = mgh
Units: Joules. h is height above a chosen reference level. Trap: only the CHANGE in height (ΔU) matters physically — the zero point of U is arbitrary and must be stated.
Formula for elastic (spring) potential energy?
Answer: U_s = ½kx²
Units: Joules. k is the spring constant (N/m), x is displacement from the spring's natural length. Trap: x is NOT the spring's total length, and U_s is always positive regardless of stretch or compression direction.
Conservation of mechanical energy (no non-conservative forces)?
Answer: KE_i + U_i = KE_f + U_f
Valid only when no friction, air resistance, or applied forces do work on the system. Trap: students forget to check whether friction/air resistance is present before applying this — if it is, mechanical energy is NOT conserved.
Energy conservation equation when non-conservative forces (like friction) act?
Answer: KE_i + U_i + W_nc = KE_f + U_f
W_nc is the work done by non-conservative forces (often negative, e.g., friction removing energy). Trap: sign convention — work done BY friction on the object is negative since friction opposes motion.
Formula for average power?
Answer: P = W/t = ΔE/t
Units: Watts (W = J/s). Trap: two objects can do the same amount of work but have very different power if the time taken differs.
Formula for instantaneous power delivered by a force?
Answer: P = Fv cos(θ)
Units: Watts. F and v are evaluated at that instant; θ is the angle between F and v. Trap: using average velocity instead of instantaneous velocity gives average power, not instantaneous power.
Formula for linear momentum?
Answer: p = mv
Units: kg·m/s. Trap: momentum is a VECTOR — direction (sign) matters and must be tracked in every momentum problem.
Formula for impulse?
Answer: J = F·Δt = Δp
Units: N·s = kg·m/s (same as momentum). Trap: use the NET force over the time interval; impulse equals the change in momentum, a vector quantity.
Impulse-Momentum Theorem (full form)?
Answer: F_net·Δt = m·v_f − m·v_i
Classic application: airbags/seatbelts/crumple zones increase Δt during a collision, which decreases the force F_net needed to produce the same change in momentum Δp.
How do you find impulse from a Force vs. time graph?
Answer: J = area under the F–t graph
Units: N·s. Trap: area below the time axis (negative force) must be subtracted, not added, to the total impulse.
Conservation of momentum for an isolated system?
Answer: Σp_i (total) = Σp_f (total), i.e. Σm_i v_i (before) = Σm_i v_i (after)
Holds when net EXTERNAL force on the system is zero. Trap: momentum is conserved in a collision even when kinetic energy is lost; always define the system to include all interacting objects.
What characterizes an ELASTIC collision, and how do you solve it?
Answer: Both momentum AND kinetic energy are conserved; solve using Σp_i=Σp_f and ΣKE_i=ΣKE_f simultaneously
Trap: 'elastic' does NOT mean the objects stick together — on the AP exam you are not required to memorize a single final-velocity formula; set up and solve the two conservation equations together.
What characterizes a PERFECTLY INELASTIC collision, and what's the final velocity formula?
Answer: Objects stick together and move with a common final velocity: v' = (m₁v₁ + m₂v₂)/(m₁ + m₂)
Units: m/s. Trap: momentum IS conserved, but kinetic energy is NOT — the 'missing' KE converts to heat, sound, and deformation.
What characterizes a general INELASTIC collision (not perfectly inelastic)?
Answer: Momentum is conserved, but some kinetic energy is lost (objects don't stick together)
Trap: distinguish 'inelastic' (some KE lost, objects separate) from 'perfectly inelastic' (maximum KE lost, objects stick) — both conserve momentum, only the perfectly inelastic case has one equation for a single final velocity.
Formula for the center of mass of a system of particles?
Answer: x_cm = (Σ m_i x_i) / (Σ m_i)
Units: meters. Trap: the center of mass can lie OUTSIDE the physical material of an object (e.g., a donut ring or an L-shaped object).
How does the center of mass move when only internal forces act (e.g., an explosion or internal collision)?
Answer: The center of mass continues moving as if nothing happened — it follows the trajectory dictated only by external forces (e.g., projectile motion under gravity)
Trap: even though fragments fly off in different directions after an explosion, the CM keeps moving along its original parabolic path since no external horizontal force acted.
Formula for torque?
Answer: τ = rF sin(θ) = r⊥F = rF⊥
Units: N·m. θ is the angle between the position vector r (from axis to point of force application) and the force F. Trap: N·m for torque is dimensionally like a Joule but is NOT energy — never convert torque units to Joules.
Condition for rotational (and full static) equilibrium?
Answer: Στ = 0 (and ΣF = 0 for complete static equilibrium)
Trap: net torque about ANY chosen pivot point is zero at equilibrium — strategically choose the pivot at the location of an unknown force to eliminate it from the torque equation.
Sign convention for torque direction?
Answer: Counterclockwise torque is positive; clockwise torque is negative (standard convention)
Trap: the magnitude of torque depends on the moment arm (perpendicular distance from axis to the line of force), which changes if you change the pivot — but equilibrium (Στ=0) holds for any valid pivot choice.
Define angular position, angular velocity, and angular acceleration.
Answer: θ (rad) = angular position; ω = Δθ/Δt (rad/s) = angular velocity; α = Δω/Δt (rad/s²) = angular acceleration
Trap: rotational kinematics equations and the v=rω, a=rα relations REQUIRE angles in radians, not degrees.
Rotational kinematics equations for constant angular acceleration?
Answer: θ = θ₀ + ω₀t + ½αt²; ω = ω₀ + αt; ω² = ω₀² + 2αΔθ
Directly analogous to linear kinematics with x→θ, v→ω, a→α. Trap: only valid when angular acceleration α is CONSTANT.
How do linear and angular quantities relate for a point on a rotating rigid body?
Answer: v = rω; a_tangential = rα; a_centripetal = v²/r = ω²r
Units: v (m/s), a (m/s²). Trap: tangential and centripetal acceleration are perpendicular components of total acceleration — a point speeding up while rotating has BOTH nonzero.
Formula for moment of inertia of a single point mass?
Answer: I = mr²
Units: kg·m². r is the distance from the AXIS OF ROTATION, not from the object's center. Trap: the same object has different I values depending on the chosen axis.
Formula for moment of inertia of a system of particles?
Answer: I = Σ m_i r_i²
Trap: spreading mass farther from the axis increases I even if total mass is unchanged — this is the physics behind a figure skater pulling arms in to spin faster (I decreases).
What are the standard moments of inertia for common shapes given on the AP formula sheet?
Answer: Solid sphere: I=(2/5)MR²; Solid cylinder/disk: I=(1/2)MR²; Hoop/ring: I=MR²; Rod about center: I=(1/12)ML²; Rod about end: I=(1/3)ML²
These are provided on the AP exam formula sheet. Trap: you must match the correct shape AND the correct axis location — a rod's I is very different about its center vs. its end.
Compare the moment of inertia of common shapes (mass M=1 kg, R or L=1 m) — which has the largest I for the same mass and size?
Answer: Hoop/ring (I=MR²=1) has the largest I; solid sphere (I=0.4) has the smallest among sphere/cylinder/hoop
A larger I means more mass is concentrated farther from the axis, making the object harder to spin up or slow down for the same torque.
Formula for rotational kinetic energy?
Answer: KE_rot = ½Iω²
Units: Joules. Trap: for an object that is both translating AND rotating (like a rolling ball), total KE = ½mv_cm² + ½Iω² — you must include BOTH terms.
Condition for rolling without slipping?
Answer: v_cm = rω
Relates the linear speed of the center of mass to the angular speed. Trap: this relation only holds when there is NO slipping; if the object slips, v_cm ≠ rω.
How does rotational KE scale with angular velocity for a fixed moment of inertia?
Answer: KE_rot grows with the square of ω (parabolic relationship), since KE_rot = ½Iω²
Example: for I = 2 kg·m², doubling ω from 1 to 2 rad/s increases KE_rot from 1 J to 4 J — a 4x increase for a 2x increase in ω.
Formula for angular momentum of a particle, and of a rigid body?
Answer: Particle: L = r×p = mvr sin(θ); Rigid body: L = Iω
Units: kg·m²/s. Trap: for a particle, r is measured from the axis/point about which you compute L, and θ is the angle between r and the velocity (or momentum) vector — analogous structure to torque.
Conservation of angular momentum — when does it apply, and what's the equation?
Answer: When net external torque is zero: L_i = L_f, i.e. I_i ω_i = I_f ω_f
Classic example: a spinning figure skater or diver pulls their arms/body in, decreasing I, which increases ω to keep L constant. Trap: L is conserved, but rotational KE is NOT conserved — KE_rot actually INCREASES as I decreases (the skater does work pulling their arms in).
Angular impulse-angular momentum theorem?
Answer: τ_net·Δt = ΔL = Iω_f − Iω_i
Units: N·m·s = kg·m²/s. This is the rotational analog of the linear impulse-momentum theorem. Trap: requires the NET external torque, and I must stay constant for this simplified form (otherwise use ΔL = I_fω_f − I_iω_i with possibly different I).
Newton's Second Law for rotation?
Answer: τ_net = Iα
Units: N·m. Direct rotational analog of F_net = ma. Trap: τ_net, I, and α must all be defined about the SAME consistent rotation axis.
What is the general free-body-diagram strategy for energy/momentum problems involving inclines, pulleys, or rotating objects?
Answer: Draw the FBD, identify which forces do work (and whether conservative or non-conservative), set a PE reference level, then apply the appropriate conservation law (energy, momentum, or angular momentum)
Trap: forces like tension in an inextensible string or the normal force on a surface typically do ZERO net work on the system (tension does equal-and-opposite work on connected objects; normal force is perpendicular to displacement) — don't mistakenly include them as doing work in your energy equation.
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