IB Physics HLRigid Body MechanicsHL onlyPaper 1 & 2~10 min read
Angular Acceleration Formula
You already know the four SUVAT equations for objects speeding up in a straight line. Here’s the good news: rotation has its own identical set. A spinning-up flywheel, a turntable slowing to a stop, a wheel accelerating from rest — all of these are solved with the very same equations, just with the angular quantities swapped in for the linear ones. If you can do linear kinematics, you can already do rotational kinematics; you just need to learn which symbol replaces which.
📘 What you need to know
The four kinematic (SUVAT) equations for uniform linear acceleration have direct rotational versions
They apply only when the angular acceleration is constant
The five variables swap: s→θ, u→ωi, v→ωf, a→α, t→t
ωf = ωi + αt
Δθ = ωit + ½αt2
ωf2 = ωi2 + 2αΔθ
Δθ = ½(ωi + ωf)t
From linear to rotational kinematics
The four kinematic equations for uniform linear acceleration will be familiar:
Linear (SUVAT) equationsv = u + ats = ut + ½at2v2 = u2 + 2ass = ½(u + v)t
Swapping each linear variable for its rotational partner gives the four rotational kinematic equations:
Each linear variable maps to a rotational partner. Only time stays the same — swap the rest and the SUVAT equations become the angular kinematic equations.
The variable swap
The whole method comes down to this correspondence. Learn which symbol replaces which, and you never have to memorise a second set of equations:
Variable
Linear
Rotational
Displacement
s
θ
Initial velocity
u
ωi
Final velocity
v
ωf
Acceleration
a
α
Time
t
t
Pick knowns 3 of 5
→ choose eq.
Match variables to rotational
→ solve
Answer in rad
WE 1
A flywheel starts at an angular velocity of 2.0 rad s−1 and accelerates uniformly at 3.0 rad s−2 for 4.0 s. Find its final angular velocity.
Step 1 — knowns are ωi, α, t; want ωf, so use ωf = ωi + αt
ωf = ωi + αt
Step 2 — substituteωf = 2.0 + (3.0 × 4.0)ωf = 14 rad s⁻¹Straight swap of the linear v = u + at — nothing new to learn.
WE 2
A wheel starts from rest and accelerates uniformly at 1.5 rad s−2 for 6.0 s. Through what angle does it turn, and how many rotations is that?
Step 1 — knowns are ωi (= 0), α, t; use Δθ = ωit + ½αt²
Δθ = ωit + ½αt²
Step 2 — substitute (ωi = 0)Δθ = 0 + ½ × 1.5 × 6.0² = 27 radStep 3 — convert to rotations (÷ 2π)27 ÷ 2π = 4.3 rotationsΔθ = 27 rad ≈ 4.3 rotationsDivide the angle in radians by 2π to count whole turns.
WE 3
A drill accelerates from rest to 50 rad s−1 while turning through 20 rad. Assuming constant angular acceleration, find α.
Step 1 — knowns are ωi (= 0), ωf, Δθ; no time, so use ωf² = ωi² + 2αΔθ
ωf² = ωi² + 2αΔθ
Step 2 — rearrange for α (ωi = 0)
α = ωf² ÷ (2Δθ)
Step 3 — substituteα = 50² ÷ (2 × 20) = 2500 ÷ 40α = 62.5 rad s⁻²This is the “no time” equation — just like v² = u² + 2as.
The trick to picking the right equation is identical to linear SUVAT: list your five quantities (θ, ωi, ωf, α, t), mark the three you know and the one you want, then choose the equation that contains those four and leaves out the one you neither know nor need. And always convert spinning rates to rad s−1 first — RPM and revolutions won’t work directly.
🛠️ Solving a rotational kinematics problem
List the five quantities: Δθ, ωi, ωf, α, t.
Convert units — angular velocity to rad s−1 (RPM ÷ 60 × 2π), angles to radians.
Mark 3 knowns + 1 wanted, and note the one you can ignore.
Pick the equation that contains your four quantities.
Substitute and solve; convert the final angle to rotations with ÷ 2π if asked.
💡 Top tips
Same skill as SUVAT. If you can pick the right linear equation, you can pick the right rotational one.
Constant α only. These equations need uniform angular acceleration — they don’t work if α changes.
RPM → rad s−1: divide by 60, then multiply by 2π before using any equation.
Rotations from radians: divide the angle by 2π (one full turn = 2π rad).
Quick recap: The four rotational kinematic equations are the linear SUVAT equations with s→θ, u→ωi, v→ωf, a→α. They apply only for constant angular acceleration. Convert rates to rad s−1 and angles to radians first, and divide an angle by 2π to count rotations.
⚠ Common mistakes
Using these equations when the angular acceleration isn’t constant
Leaving angular velocity in RPM or rev s−1 instead of converting to rad s−1
Forgetting to divide by 2π when a question asks for the number of rotations
Dropping the negative sign on α when something is decelerating (slowing to a stop)
Picking an equation that doesn’t contain your three knowns and one unknown
You can now describe how a rotating body’s motion changes over time. The missing piece is what causes that angular acceleration in the first place — and just as force is spread over mass in F = ma, torque is spread over a quantity called the moment of inertia. That’s the next building block.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
