IB Physics HLRigid Body MechanicsHL onlyPaper 1 & 2~11 min read
Angular Displacement, Velocity & Acceleration
Everything you know about straight-line motion — displacement, velocity, acceleration — has a rotational twin. A spinning wheel doesn’t travel in a line, but it still turns through an angle, at some rate, and that rate can change. These three “angular” quantities describe rotating bodies, and the beautiful part is that each one links back to its linear cousin through a single factor: the radius. Learn those links and rotational motion stops feeling foreign — it’s just circular geometry wrapped around the motion you already understand.
📘 What you need to know
Angular displacement Δθ is the change in angle of a rotating body, measured in radians
Linear and angular displacement are linked by s = rΔθ
Angular velocityω is the rate of change of angular displacement, in rad s−1
Linear and angular velocity are linked by v = rω
ω = v / r = 2πf = 2π / T
Angular accelerationα is the rate of change of angular velocity, in rad s−2
Linear and angular acceleration are linked by a = rα
Angular displacement
Angular displacement is the change in angle through which a rigid body has rotated relative to a fixed point. It’s measured in radians — the natural unit for rotation, where one radian is the angle that makes the arc length equal to the radius.
Linear and angular displacements = rΔθ
where s is the arc length (the linear distance travelled along the circular path) in metres (m), r is the radius of the circular path in metres (m), and Δθ is the angular displacement in radians (rad). Rearranged, this gives the very definition of an angle in radians: θ = s / r, the arc length divided by the radius.
An angle in radians is the arc length s divided by the radius r. Multiplying back gives the linear distance s = rΔθ.
WE 1
A point on the edge of a wheel of radius 0.35 m rotates through an angular displacement of 4.0 rad. Calculate the linear distance (arc length) it travels.
Step 1 — use s = rΔθ
s = r × Δθ
Step 2 — substitute (angle already in radians)s = 0.35 × 4.0s = 1.4 mThe angle must be in radians for this to work — that’s the whole point of the radian.
Angular velocity
Angular velocity ω is the rate of change of angular displacement with respect to time — how fast the body is turning. It’s measured in radians per second (rad s−1):
Angular velocityω = Δθ / Δt
The linear speed v of a point on the rotating body is related to the angular velocity by the radius:
Linear and angular velocityv = rω
Because one complete rotation is an angular displacement of 2π radians, angular velocity can also be written in terms of the frequency f or the time period T:
Angular velocity, frequency and periodω = v / r = 2πf = 2π / T
WE 2
A fan spins at 1200 revolutions per minute (rpm). Calculate its angular velocity, and the linear speed of a point on a blade tip 0.25 m from the axis.
Step 1 — convert rpm to rev per second, then use ω = 2πf
f = 1200 ÷ 60 = 20 rev s⁻¹
ω = 2π × 20 = 125.7 rad s⁻¹Step 2 — find the tip speed with v = rωv = 0.25 × 125.7ω = 126 rad s⁻¹, v = 31.4 m s⁻¹Every point turns at the same ω, but points further out (bigger r) move faster.
Here’s the intuition that ties it all together: every point on a rigid rotating body shares the same angular velocity — they all sweep the same angle in the same time. But their linear speeds differ, because v = rω means points further from the axis cover more distance. It’s why the outer edge of a merry-go-round whips past faster than the middle, even though the whole thing turns as one.
Angular acceleration
Angular acceleration α is the rate of change of angular velocity with time — how quickly the spin is speeding up or slowing down. It’s measured in radians per second squared (rad s−2):
Angular accelerationα = Δω / Δt
And, completing the pattern, linear acceleration a (specifically the tangential acceleration of a point) links to angular acceleration through the radius:
Linear and angular accelerationa = rα
WE 3
A disc speeds up from an angular velocity of 5.0 rad s−1 to 20.0 rad s−1 in 3.0 s. Calculate its angular acceleration, and the tangential linear acceleration of a point 0.10 m from the axis.
Step 1 — use α = Δω ÷ Δtα = (20.0 − 5.0) ÷ 3.0 = 15 ÷ 3.0 = 5.0 rad s⁻²Step 2 — find the linear acceleration with a = rαa = 0.10 × 5.0α = 5.0 rad s⁻², a = 0.50 m s⁻²The same radius factor links all three pairs of quantities — displacement, velocity and acceleration.
Graphs of rotational motion
Rotational-motion graphs behave exactly like their linear counterparts — the same gradient and area rules apply, just with angular quantities. Read them the same way you read displacement, velocity and acceleration graphs.
The gradient of the θ–t graph gives ω; the gradient of the ω–t graph gives α. Areas work in reverse: the area under ω–t gives θ, and the area under α–t gives ω.
Angular velocity = gradient of the angular displacement–time graph, and the area under the angular acceleration–time graph
Angular displacement = area under the angular velocity–time graph
Angular acceleration = gradient of the angular velocity–time graph
Summary of linear and angular variables
Each rotational quantity is its linear partner divided by the radius (or multiplied, going the other way). This table is worth committing to memory — it turns every rotational problem into a familiar linear one:
Variable
Linear
Angular
Displacement
s = rθ
θ = s / r
Velocity
v = rω
ω = v / r
Acceleration
a = rα
α = a / r
Linear s, v, a
÷ r →
Angular θ, ω, α
← × r
back to linear
🛠️ Switching between linear and angular
Spot which quantity you have — displacement, velocity or acceleration.
Angle must be in radians — convert degrees or revolutions first (1 rev = 2π rad).
Multiply by r to go from angular to linear (s = rθ, v = rω, a = rα).
Divide by r to go from linear to angular.
For spinning rates, use ω = 2πf = 2π / T to bring in frequency or period.
💡 Top tips
Always radians. Every angular equation here needs the angle in radians, never degrees.
One factor rules them all. The same radius r links displacement, velocity and acceleration — learn one, learn all three.
rpm to rad s−1: divide by 60 to get rev s−1, then multiply by 2π.
Same graph rules. Gradients and areas mean exactly what they do for linear motion.
Quick recap: Angular displacement (rad), velocity (rad s−1) and acceleration (rad s−2) mirror their linear partners, each linked by the radius: s = rθ, v = rω, a = rα. Angular velocity is also 2πf = 2π / T, and rotational graphs follow the same gradient and area rules as linear ones.
⚠ Common mistakes
Using degrees instead of radians in the angular equations
Confusing angular acceleration (α) with centripetal acceleration — they’re not the same thing
Forgetting to convert rpm or rev s−1 into rad s−1 before using ω
Assuming all points on a rotating body have the same linear speed — only ω is shared
Reading a rotational graph differently from a linear one — the gradient and area rules are identical
Now you have the angular quantities and how they connect to linear motion. The next step is to describe changing rotation with equations — and just as linear motion has its four SUVAT equations, rotation has its own set of kinematic equations. That’s the angular acceleration formula set, coming up next.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
