🔄Rotational Kinematics
Angular Variables
Every linear kinematic quantity has a rotational analog:
| Linear | Rotational | |--------|-----------| | position x (m) | angle θ (rad) | | velocity v (m/s) | angular velocity ω (rad/s) | | acceleration a (m/s²) | angular acceleration α (rad/s²) |
The rotational kinematic equations are identical in form to the linear ones — just replace linear quantities with angular:
v → ω, a → α, x → θ
📐 Rotational Kinematic Equations
For constant angular acceleration (compare to linear kinematics):
ω = ω_0 + α tθ = ω_0 t + (1)/(2)α t²ω² = ω_0² + 2αθv = rω, a_t = rα✏️ Worked Example
Problem: A wheel starts from rest and reaches 120 rpm in 4 seconds. Find the angular acceleration.
⚠️ Common Mistakes
Misconception: Rotational kinematic equations can be used without converting rpm to rad/s.
✓ Correct thinking: All rotational kinematic equations require angular quantities in SI units: θ in radians, ω in rad/s, α in rad/s².
Why: RPM is not an SI unit. Mixing units (e.g., using rpm directly in ω = ω₀ + αt) gives wrong answers.
Misconception: The linear (tangential) speed v is the same for all points on a rotating object.
✓ Correct thinking: ω is the same for all points, but v = rω varies with radius. Points farther from the axis move faster.
Why: All parts of a rigid body complete one revolution in the same time, but outer points travel a larger circumference — so v is larger there.
Misconception: Centripetal acceleration and tangential acceleration are the same thing.
✓ Correct thinking: They are perpendicular to each other. a_t = rα (along the direction of motion) and a_c = rω² (toward the center).
Why: a_t changes the speed; a_c changes the direction. A point can have both simultaneously if the object is both speeding up and curving.
📝 Practice Problems
Try these problems. Check your answer when ready.
Convert 240 rpm to rad/s.
A wheel accelerates from rest at α = 2 rad/s². How long does it take to reach ω = 12 rad/s?
ω = ω_0 + α tA disk (R = 0.3 m) starts from rest and reaches 10 rad/s in 5 s with constant α. Find (a) α, (b) total angle rotated, (c) the tangential speed of the rim at t = 5 s.
A grinding wheel rotating at 1200 rpm is switched off and decelerates uniformly to rest in 30 s. How many revolutions does it make before stopping?
A point on the rim of a wheel (R = 0.5 m) has a centripetal acceleration of 50 m/s². Find ω and the tangential speed.
a_c = rω²A flywheel rotates at 60 rad/s. A braking torque decelerates it at α = −3 rad/s². (a) When does it stop? (b) How many radians does it travel? (c) What is the total angle in revolutions?
A motor accelerates a rotor from rest to 3000 rpm in 10 s with constant angular acceleration. Find (a) α in rad/s², (b) total angle in radians, and (c) the tangential acceleration of a point 0.4 m from the axis.
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