Kinematics of Rotation
& System Dynamics
The equations of motion for rotating bodies — built by analogy, verified by intuition.
The Linear ↔ Rotational Analogy
This analogy is the fastest way to write rotational equations. If you've solved 100 linear kinematics problems, you've also solved 100 rotational ones — just replace symbols. Master this table and save 30 seconds per question.
| Linear | Symbol | Rotational | Symbol | Connection |
|---|---|---|---|---|
| Displacement | x | Angular Displacement | θ | x = rθ |
| Velocity | v | Angular Velocity | ω | v = rω |
| Acceleration | a | Angular Acceleration | α | aₜ = rα |
| Mass | m | Moment of Inertia | I = Σmr² | — |
| Force (F = ma) | F | Torque (τ = Iα) | τ | — |
| Linear Momentum (p = mv) | p | Angular Momentum (L = Iω) | L | — |
| KE = ½mv² | — | KE = ½Iω² | — | — |
| Work = Fs | — | Work = τθ | — | — |
| Power = Fv | — | Power = τω | — | — |
| Impulse = FΔt | — | Angular Impulse = τΔt | — | ΔL = τΔt |
Rotational Kinematic Equations
For constant angular acceleration (α = constant):
These are the SAME equations as linear, just with different symbols. When a question gives θ, ω₀, α — reach for equation 3. When it asks for time → equation 1 or 2. When time isn't given → equation 3. The selection logic is identical to linear kinematics.
θ in these equations is in RADIANS, not degrees. Angular velocity ω must be in rad/s. If frequency f is given, convert: ω = 2πf. Forgetting this unit conversion is a classic careless error.
Variable Angular Acceleration (calculus approach)
When α = f(t) or α = f(ω), use calculus. Example: α = kω (where k is constant). Then dω/dt = kω → ∫dω/ω = ∫k dt → ln(ω/ω₀) = kt → ω = ω₀eᵏᵗ. This is exponential growth of angular velocity. JEE 2019 had a variant of this type.
Dynamics of a System of Particles
In a system of particles, internal forces are forces between particles within the system. By Newton's 3rd law, they occur in action-reaction pairs and cancel out.
External forces are forces on the system from outside. They affect the total momentum and COM acceleration.
Even if individual particles experience complex internal forces, the COM moves AS IF it's a single particle of mass M under external force F_ext. This is the power of the COM framework.
Application examples:
- Explosion of shell mid-air → COM continues parabolic path
- Rocket propulsion (variable mass system)
- Gun recoil
- Man jumping from boat
When a shell explodes mid-air, the COM continues on the same parabolic path. Students incorrectly think the explosion changes the COM trajectory. It doesn't — there's no external force change (gravity was already acting before explosion).
where v_rel = velocity of exhaust gas relative to rocket, dm/dt = rate of mass ejection (negative, since mass is decreasing).
Rocket problems in JEE Advanced often ask for velocity at a given time or when mass reduces to m₀/e. Use: Δv = v_rel × ln(M₀/M). The "Tsiolkovsky rocket equation." This derivation is expected in JEE Advanced answers.
A rigid body can undergo both translation (of COM) and rotation (about COM) simultaneously.
L_orbital = angular momentum due to COM motion (like Earth orbiting Sun). L_spin = angular momentum due to rotation about own axis (like Earth spinning). Both are measured from the same origin. This is crucial for satellite + planet problems in JEE.
🧮 Rotational Kinematics Calculator
Find ω, θ using kinematic equations
τ = F × r × sin(θ)
v = √[2gh/(1+k²/R²)]
System-Level Thinking
A man runs on a frictionless surface: he can't move! With friction, force by ground on feet accelerates the system. The internal forces within his body push against the ground via friction. No friction → no external force → COM at rest.
Explosive forces are internal. COM continues on original trajectory. Find positions/velocities of pieces using momentum conservation in each direction independently.
Gun fires bullet: system initially at rest. p_bullet + p_gun = 0. Bullet goes forward, gun recoils backward. Velocities are inversely proportional to masses.
Man walks on a floating boat. No external horizontal force → COM doesn't move. Boat moves backward as man moves forward. Classic JEE problem.
Man-boat and explosion problems appear frequently in JEE Main (2015, 2017, 2019, 2021). The template is always: (1) Identify if external force = 0. (2) COM doesn't move. (3) Write Σmᵢxᵢ = M×X_cm = constant. (4) Differentiate or use displacement form. Done in <90 seconds.