Rotational Motion — Complete Formula

Angular Variables

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Angular Displacement	[\theta = \dfrac{s}{r}]	[\theta] = angular displacement, [s] = arc length, [r] = radius	rad	Analog of linear displacement
Angular Velocity	[\omega = \dfrac{d\theta}{dt}]	[\omega] = angular velocity, [t] = time	rad/s	Same for all points in rigid body ⭐
Angular Acceleration	[\alpha = \dfrac{d\omega}{dt}]	[\alpha] = angular acceleration	rad/s²	Rate of change of angular velocity

💡 Memory Hint:
Linear ↔ Angular:
[s → \theta], [v → \omega], [a → \alpha]

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
First Equation	[\omega = \omega_0 + \alpha t]	[\omega_0] = initial angular velocity	rad/s	Direct analog of [v = u + at] ⭐
Second Equation	[\theta = \omega_0 t + \dfrac{1}{2}\alpha t^2]	—	rad	Use when time is given
Third Equation	[\omega^2 = \omega_0^2 + 2\alpha \theta]	—	rad²/s²	Use when time not given ⭐

💡 Memory Hint:
Same as linear motion → just replace variables

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Linear Velocity	[v = r\omega]	[v] = linear velocity	m/s	Tangential velocity ⭐
Tangential Acceleration	[a_t = r\alpha]	[a_t] = tangential acceleration	m/s²	Due to change in speed
Centripetal Acceleration	[a_c = \dfrac{v^2}{r} = r\omega^2]	—	m/s²	Towards center ⭐

💡 Memory Hint:
Multiply by [r] → convert angular → linear

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Torque	[\vec{\tau} = \vec{r} \times \vec{F}]	[\tau] = torque, [r] = position vector, [F] = force	N·m	Rotational analog of force ⭐
Magnitude	[\tau = rF\sin\theta]	[\theta] = angle between r and F	N·m	Only perpendicular component works ⚠️
Newton’s Second Law (Rotation)	[\tau = I\alpha]	[I] = moment of inertia	N·m	Most important ⭐

💡 Memory Hint:
Torque = force × perpendicular distance

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Definition	[I = \sum mr^2]	[m] = mass, [r] = distance from axis	kg·m²	Rotational inertia ⭐
Continuous Form	[I = \int r^2 dm]	[dm] = small mass element	kg·m²	For continuous bodies
Parallel Axis Theorem	[I = I_{cm} + Md^2]	[I_{cm}] = about centre, [d] = distance	kg·m²	Very important ⭐
Perpendicular Axis Theorem	[I_z = I_x + I_y]	—	kg·m²	Only for planar bodies ⚠️

💡 Memory Hint:
Mass farther from axis → higher inertia

Body	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Ring (about center)	[I = MR^2]	[M] = mass, [R] = radius	kg·m²	All mass at same distance
Disc (about center)	[I = \dfrac{1}{2}MR^2]	—	kg·m²	Most common ⭐
Rod (center)	[I = \dfrac{1}{12}ML^2]	[L] = length	kg·m²	Axis through centre
Rod (end)	[I = \dfrac{1}{3}ML^2]	—	kg·m²	Use parallel axis ⭐
Sphere (solid)	[I = \dfrac{2}{5}MR^2]	—	kg·m²	Standard result
Sphere (hollow)	[I = \dfrac{2}{3}MR^2]	—	kg·m²	Higher than solid ⚠️

💡 Memory Hint:
Hollow bodies → larger MI than solid

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Angular Momentum	[\vec{L} = \vec{r} \times \vec{p}]	[L] = angular momentum, [p] = linear momentum	kg·m²/s	Vector quantity
Rigid Body	[L = I\omega]	—	kg·m²/s	Most used form ⭐
Relation with Torque	[\tau = \dfrac{dL}{dt}]	—	N·m	Analog of F = dp/dt ⭐

💡 Memory Hint:
Angular momentum = rotational momentum

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Work Done	[W = \tau \theta]	[\theta] = angular displacement	J	Rotational work
Kinetic Energy	[K = \dfrac{1}{2}I\omega^2]	—	J	Rotational KE ⭐
Power	[P = \tau \omega]	—	W	Rotational power ⭐

💡 Memory Hint:
Replace [F → τ], [v → ω]

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Condition for Rolling	[v = R\omega]	[R] = radius	m/s	No slipping ⭐
Total KE (Rolling)	[K = \dfrac{1}{2}mv^2 + \dfrac{1}{2}I\omega^2]	—	J	Translation + rotation ⭐
Acceleration (Incline)	[a = \dfrac{g\sin\theta}{1 + \dfrac{I}{mR^2}}]	—	m/s²	Very important ⭐

💡 Memory Hint:
Rolling = translation + rotation combined

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