On this page
If you can solve a car braking problem, you already know rotational kinematics. The equations are the same with new symbols. Yet students lose marks here because of units: revolutions per minute, radians and revolutions get mixed in one line. This post sets out the equations, shows how to convert units safely, and works through three problems, including one where it is quicker not to calculate α at all.
What does the syllabus include?
The Syllabus for JEE (Main) 2026 lists "rigid body rotation and equations of rotational motion, comparison of linear and rotational motions" in Unit 5. The equations are in Section 6.10 of the NCERT Class 11 chapter on systems of particles and rotational motion. Check the syllabus for your own exam year.
What are the equations for constant angular acceleration?
For rotation about a fixed axis, angular displacement θ, angular velocity ω and angular acceleration α play the parts of x, v and a. With constant α, NCERT gives:
ω = ω₀ + αt (like v = v₀ + at)
θ = ω₀t + ½αt² (like x = v₀t + ½at²)
ω² = ω₀² + 2αθ (like v² = v₀² + 2ax)
One more is often handy: θ = ½(ω₀ + ω)t, the average angular velocity times time.
Points on the body link the angular and linear quantities. A point at distance r from the axis has speed v = ωr, tangential acceleration a_t = αr, and centripetal acceleration ω²r directed towards the axis.
OpenStax University Physics shows how realistic these numbers are with a fishing reel given an angular acceleration of 110 rad/s² for 2.00 s: it ends up spinning at 220 rad/s, about 2100 rpm.
How do you convert rpm and revolutions?
rpm to rad/s: multiply by 2π/60. So 1200 rpm = 40π rad/s.
Radians to revolutions: divide by 2π.
NCERT's Example 6.11 uses these: a motor wheel goes from 1200 rpm to 3120 rpm in 16 s. Converting gives 40π and 104π rad/s, so α = 64π ÷ 16 = 4π rad/s², and the wheel turns through 1152π rad, which is 576 revolutions.
Worked example 1: a ceiling fan switched off
A fan turning at 300 rpm is switched off and slows uniformly to rest in 60 s. Find its angular acceleration, the number of turns it makes while stopping, and the accelerations of a blade tip 0.60 m from the axis just after switch-off.
ω₀ in rad/s: 300 × 2π/60 = 10π ≈ 31.4 rad/s.
α: 0 = 10π + α(60), so α = -π/6 ≈ -0.524 rad/s². The minus sign means it slows down.
Angle: θ = ½(ω₀ + ω)t = ½(10π + 0)(60) = 300π rad.
Turns: 300π ÷ 2π = 150 revolutions.
Blade tip: speed v = ωr = 31.4 × 0.60 ≈ 18.8 m/s. Tangential acceleration a_t = αr = 0.524 × 0.60 ≈ 0.31 m/s² (slowing). Centripetal acceleration ω²r = (31.4)² × 0.60 ≈ 592 m/s².
Notice how small a_t is compared with the centripetal acceleration. Questions on the total acceleration of a point need both, at right angles.
Worked example 2: turns in the next two seconds
A wheel starts from rest with constant α and makes 10 revolutions in the first 2 s. How many does it make in the next 2 s?
From rest, θ = ½αt², so θ is proportional to t².
In 4 s: doubling t multiplies θ by 4, giving 40 revolutions.
In the next 2 s: 40 - 10 = 30 revolutions.
You do not need α, but for the record: 10 revolutions = 20π rad, so α = 2θ/t² = 40π ÷ 4 = 10π ≈ 31.4 rad/s².
Worked example 3: using ω² = ω₀² + 2αθ
A flywheel slows uniformly from 120 rad/s to 80 rad/s while turning through 50 revolutions. How many more revolutions does it make before stopping?
θ in radians: 50 × 2π = 100π rad.
α: 80² = 120² + 2α(100π), so α = (6400 - 14,400) ÷ 200π ≈ -12.7 rad/s².
To stop from 80 rad/s: 0 = 80² + 2αθ', so θ' = 6400 ÷ (2 × 12.7) ≈ 251 rad ≈ 40 revolutions.
Quick check: with constant α, θ is proportional to ω₀² - ω². From 120 to 0 is proportional to 14,400; from 80 to 0, to 6400. The ratio 14,400 : 6400 is 9 : 4, so if the whole stop takes 90 revolutions, the last part takes 40 and the first part 50. It matches.
Which mistakes cost marks?
Putting rpm into the equations. Convert to rad/s first.
Reporting radians as revolutions (or the reverse).
Dropping the sign of α when a body slows down.
Using these equations when α changes with time. If α = kt, integrate instead: ω = ω₀ + ½kt².
Confusing a_t with centripetal acceleration. Only a_t = αr depends on α.
A practice routine for this topic
Write the three equations beside their linear twins from memory before each session.
Do one conversion drill: five speeds from rpm to rad/s and five angles from radians to revolutions.
Solve one problem three ways where possible, as in Example 3, and check the answers agree.
Add one rotational dynamics problem once kinematics is steady, such as a string unwinding from a cylinder.
If you are not sure which equation fits, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. Our list of rotational motion mistakes covers the dynamics side.
Start now: redo Example 1 for a fan at 450 rpm that stops in 60 s, and check that it makes 225 turns.
