Rotational Motion
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Angular Kinematics & Relations
Angular Velocity (ω)
ω = dθ/dt
ωₐᵥ = Δθ/Δt
ωₐᵥ = Δθ/Δt
Unit: rad/s. Vector direction by right-hand rule.
Angular Acceleration (α)
α = dω/dt
αₐᵥ = Δω/Δt
αₐᵥ = Δω/Δt
Unit: rad/s².
Linear-Angular Relation
v = ωr
aₜ = αr
aₜ = αr
v is tangential velocity, aₜ is tangential acceleration.
Centripetal Acceleration
a꜀ = v²/r = ω²r
Always directed towards the center of the circle.
Vector Relations
v⃗ = ω⃗ × r⃗
a⃗ = α⃗ × r⃗ + ω⃗ × v⃗
a⃗ = α⃗ × r⃗ + ω⃗ × v⃗
a⃗ is total acceleration (tangential + centripetal).
Constant Angular Acceleration Equations
First Equation
ω = ω₀ + αt
Analogous to v = u + at.
Second Equation
θ = ω₀t + ½αt²
Analogous to s = ut + ½at².
Third Equation
ω² = ω₀² + 2αθ
Analogous to v² = u² + 2as.
Displacement in nᵗʰ sec
θₙ = ω₀ + (α/2)(2n-1)
Useful for specific time interval questions.
Moment of Inertia (I) & Rotational KE
Moment of Inertia (I)
I = Σmᵢrᵢ² = ∫r²dm
Rotational analog of mass. Depends on mass distribution and axis.
Rotational Kinetic Energy
K.E.ᵣₒₜ = ½Iω²
Energy due to rotation. Analogous to ½mv².
Parallel Axis Theorem
I = I꜀ₘ + Md²
I꜀ₘ is MOI about center of mass. `d` is perp. distance between axes.
Perpendicular Axis Theorem
I₂ = Iₓ + Iᵧ
Only for 2D planar bodies. x, y axes in plane, z is perp.
Radius of Gyration (k)
I = Mk² => k = √(I/M)
Effective distance of mass from axis of rotation.
Torque (τ) & Angular Momentum (L)
Torque (τ)
τ⃗ = r⃗ × F⃗
|τ| = rFsinθ
|τ| = rFsinθ
Rotational analog of Force. Causes angular acceleration.
Newton's 2nd Law (Rotation)
τₑₓₜ = Iα
Directly relates net external torque to angular acceleration.
Angular Momentum (L)
L⃗ = r⃗ × p⃗ (particle)
L = Iω (rigid body)
L = Iω (rigid body)
Rotational analog of linear momentum (p).
Conservation of L
If τₑₓₜ = 0, then dL/dt = 0
L = constant => I₁ω₁ = I₂ω₂
L = constant => I₁ω₁ = I₂ω₂
Key principle for problems like spinning skaters, planetary motion.
Rolling Motion (Without Slipping)
Condition for Pure Rolling
v꜀ₘ = Rω
a꜀ₘ = Rα
a꜀ₘ = Rα
Velocity of the point of contact with the ground is zero.
Total Kinetic Energy
K.E.ₜₒₜ = K.E.ₜᵣₐₙₛ + K.E.ᵣₒₜ
= ½mv꜀ₘ² + ½I꜀ₘω²
= ½mv꜀ₘ² + ½I꜀ₘω²
Sum of translational and rotational kinetic energies.
Total KE (Shortcut)
K.E.ₜₒₜ = ½mv꜀ₘ² (1 + k²/R²)
k is the radius of gyration. Very useful for quick calculations.
Acceleration on Inclined Plane
a = g sinθ / (1 + I/mR²)
= g sinθ / (1 + k²/R²)
= g sinθ / (1 + k²/R²)
Memorize this! It's a very common BITSAT question.
BITSAT Exam Strategy
Memorize I for common shapes: Ring (MR²), Disc (½MR²), Solid Sphere (²/₅MR²), Hollow Sphere (²/₃MR²), Rod (¹/₁₂ML² or ¹/₃ML²).
For conservation of L problems (I₁ω₁ = I₂ω₂), clearly identify the 'before' and 'after' states. E.g., a person walking on a turntable.
In rolling problems, the factor β = (1 + k²/R²) appears everywhere. Calculate it first to save time.
Use dimensional analysis to eliminate options for Torque (ML²T⁻²) and Angular Momentum (ML²T⁻¹).
For objects racing down an incline, smaller k²/R² ratio wins. Solid Sphere > Disc > Hollow Sphere > Ring.
Linear vs. Rotational Motion Analogy
| Linear Motion | Rotational Motion | Relation |
|---|---|---|
Displacement: s | Angular Displacement: θ | s = rθ |
Velocity: v | Angular Velocity: ω | v = rω |
Acceleration: a | Angular Acceleration: α | aₜ = rα |
Mass (Inertia): m | Moment of Inertia: I | I = Σmr² |
Force: F | Torque: τ | τ = r × F |
Momentum: p = mv | Angular Momentum: L = Iω | L = r × p |
Newton's 2nd Law: F = ma | Newton's 2nd Law: τ = Iα | Analogous Laws |
Kinetic Energy: ½mv² | Rotational K.E.: ½Iω² | Energy Forms |