θ = angular position (radians)
ω = dθ/dt = angular velocity (rad/s)
α = dω/dt = angular acceleration (rad/s²)
ω = ω₀ + αt
θ = θ₀ + ω₀t + ½αt²
ω² = ω₀² + 2α(θ - θ₀)
θ = θ₀ + ½(ω + ω₀)t
A wheel starts from rest and accelerates at 2 rad/s². What is its angular velocity after 3 seconds?
Solution:
ω = ω₀ + αt
ω = 0 + (2 rad/s²)(3 s) = 6 rad/s
τ = r × F = rF sin θ
where:
Key Points:
I = Σmᵢrᵢ² (discrete)
I = ∫r²dm (continuous)
Στ = Iα
Similar to ΣF = ma for linear motion
| Linear | Rotational |
|---|---|
| Force (F) | Torque (τ) |
| Mass (m) | Moment of Inertia (I) |
| Linear acceleration (a) | Angular acceleration (α) |
W = ∫τ dθ
Power: P = τω
KE_rot = ½Iω²
A 2 kg disk with radius 0.3 m spins at 10 rad/s. Calculate its rotational kinetic energy.
Solution:
I = ½mr² = ½(2 kg)(0.3 m)² = 0.09 kg⋅m²
KE = ½Iω² = ½(0.09)(10)² = 4.5 J
L = r × p = mvr sin θ (particle)
L = Iω (rigid body)
When net external torque is zero:
L = constant
Iω = constant
v = Rω (no slipping)
Total Energy = ½mv² + ½Iω²
a = (mg sin θ)/(m + I/R²)
A solid sphere rolls down a 30° incline. Find its acceleration.
Solution:
I = ⅖mR²
a = (g sin θ)/(1 + ⅖) = (9.81 sin 30°)/1.4 = 3.5 m/s²
Ω = mgR/(Iω)
where:
Steps for Rotational Dynamics Problems:
AP Exam Tips:
