Constants (always given on the exam) G = 6.67×10⁻¹¹ N·m²/kg² · g = 9.8 m/s²Exam conventions: reference frames are inertial, air resistance is negligible, and springs/strings are ideal — all unless a problem states otherwise.
Calculus Toolkit (Unit 0) 11 equations d/dx(xⁿ) = nxⁿ⁻¹Power Rule for Derivatives ▼
∫xⁿ dx = 1/(n+1) xⁿ⁺¹, n ≠ −1Power Rule for Integrals ▼
df/dx = (df/du)(du/dx)Chain Rule ▼
d/dx[sin(ax)] = a cos(ax)Derivative of Sine ▼
d/dx[cos(ax)] = −a sin(ax)Derivative of Cosine ▼
d/dx(eᵃˣ) = aeᵃˣDerivative of an Exponential ▼
d/dx(ln ax) = 1/xDerivative of a Natural Log ▼
∫eᵃˣ dx = 1/a eᵃˣIntegral of an Exponential ▼
∫dx/(x+a) = ln|x+a|Integral of a Reciprocal ▼
∫cos(ax)dx = 1/a sin(ax)Integral of Cosine ▼
∫sin(ax)dx = −1/a cos(ax)Integral of Sine ▼
vₓ = vₓ₀ + aₓtVelocity-Time Equation (Constant Acceleration) ▼
x = x₀ + vₓ₀t + ½aₓt²Position-Time Equation (Constant Acceleration) ▼
vₓ² = vₓ₀² + 2aₓ(x − x₀)Velocity-Position Equation ▼
Δx = ∫v(t) dtDisplacement as the Integral of Velocity ▼
Δvₓ = ∫a(t) dtChange in Velocity as the Integral of Acceleration ▼
Force and Translational Dynamics 9 equations x̄_cm = Σmᵢxᵢ / ΣmᵢCenter of Mass Position (Discrete) ▼
r̄_cm = ∫r dm / ∫dmCenter of Mass Position (Continuous) ▼
λ = dm/dℓLinear Mass Density ▼
ā_sys = ΣF̄ / m_sys = F̄_net / m_sysSystem (Center-of-Mass) Acceleration ▼
|F̄_g| = Gm₁m₂ / r²Newton's Law of Universal Gravitation ▼
|F̄_f| ≤ |μF̄_N|Friction Force ▼
F̄_s = −kΔx̄Spring Force (Hooke's Law) ▼
aₒ = v²/r = rω²Centripetal Acceleration ▼
T = 1/fPeriod-Frequency Relationship ▼
Work, Energy, and Power 10 equations K = ½mv²Translational Kinetic Energy ▼
W = ∫[a→b] F̄·dr̄Work as a Line Integral ▼
ΔK = ΣWᵢ = ΣF∥ᵢdᵢWork-Energy Theorem ▼
ΔU = −∫[a→b] F̄_cf(r)·dr̄Potential Energy (General Definition) ▼
Fₓ = −dU(x)/dxForce from Potential Energy ▼
Uₛ = ½k(Δx)²Spring (Elastic) Potential Energy ▼
U_G = −Gm₁m₂/rGravitational Potential Energy (General) ▼
ΔU_g = mgΔyGravitational PE (Near Earth's Surface) ▼
P_avg = W/Δt = ΔE/ΔtAverage Power ▼
P_inst = dW/dtInstantaneous Power ▼
Linear Momentum 4 equations p̄ = mv̄Linear Momentum ▼
F̄_net = dp̄/dtNewton's Second Law (General Form) ▼
J̄ = ∫[t₁→t₂] F̄_net(t) dt = Δp̄Impulse-Momentum Theorem ▼
v̄_cm = Σp̄ᵢ/Σmᵢ = Σmᵢv̄ᵢ/ΣmᵢCenter of Mass Velocity ▼
Torque and Rotational Dynamics 12 equations ω = dθ/dtAngular Velocity (Definition) ▼
α = dω/dtAngular Acceleration (Definition) ▼
ω = ω₀ + αtRotational Velocity-Time Equation ▼
θ = θ₀ + ω₀t + ½αt²Rotational Position-Time Equation ▼
ω² = ω₀² + 2α(θ − θ₀)Rotational Velocity-Position Equation ▼
v = rωTangential Speed ▼
a_T = rαTangential Acceleration ▼
τ̄ = r̄ × F̄Torque ▼
I_tot = Σmᵢrᵢ²Rotational Inertia (Point Masses) ▼
I = ∫r² dmRotational Inertia (Continuous) ▼
I′ = I_cm + Md²Parallel Axis Theorem ▼
α_sys = Στ/I_sys = τ_net/I_sysNewton's Second Law for Rotation ▼
Energy and Momentum of Rotating Systems 5 equations K_rot = ½Iω²Rotational Kinetic Energy ▼
W = ∫τ dθRotational Work ▼
L̄ = r̄ × p̄ = Iω̄Angular Momentum ▼
ΔL = ∫τ dtAngular Impulse-Momentum Theorem ▼
Δx_cm = rΔθRolling Without Slipping (Displacement) ▼
T = 2π/ω = 1/fPeriod-Frequency-Angular Frequency Relationship ▼
T_s = 2π√(m/k)Period of a Spring-Mass Oscillator ▼
T_p = 2π√(ℓ/g)Period of a Simple Pendulum ▼
T_phys = 2π√(I/mgd)Period of a Physical Pendulum ▼
x = x_max cos(ωt + φ)Displacement in SHM ▼
Unit 0
Calculus Primer — supplementary
Unit 2
Force and Translational Dynamics
Unit 3
Work, Energy, and Power
Unit 5
Torque and Rotational Dynamics
Unit 6
Energy and Momentum of Rotating Systems
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