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System of Particles and Rotational Motion cheat sheet

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JEE · Physics

Cheat sheet

Class 11 · Chapter 6System of Particles and Rotational Motion

Rotational twins: F ↔ τ, m ↔ I, p ↔ L, v ↔ ω.

1. Centre of Mass

\vec R_{cm} = \dfrac{\sum m_i\vec r_i}{M}

CM moves as if all mass and external force were at that point. Internal forces don't change its motion.

2. Torque & Angular Momentum

\vec\tau = \vec r\times\vec F

\vec L = \vec r\times\vec p

\vec\tau = \dfrac{d\vec L}{dt}

L = I\omega

Conservation: if $\tau_{ext} = 0$ , $I\omega$ = constant (spinning skater pulls arms in → spins faster).

3. Linear vs Rotational

Linear	Rotational
x	θ
v	$\omega$
m	$I = \sum mr^2$
F = ma	$\tau = I\alpha$
$\tfrac12 mv^2$	$\tfrac12 I\omega^2$
P = Fv	$P = \tau\omega$

4. Moments of Inertia

Body (axis)	I
Ring (centre, ⊥)	$MR^2$
Disc (centre, ⊥)	$\tfrac12 MR^2$
Rod (centre, ⊥)	$\tfrac{1}{12}ML^2$
Solid sphere (diameter)	$\tfrac25 MR^2$

5. Equilibrium

Translational: net force = 0.
Rotational: net torque = 0.
Couple: equal, opposite, non-collinear forces — produces only rotation.

6. Key Points

Rolling KE = ½mv²(1 + k²/R²).
Radius of gyration: I = Mk².
Angular momentum is conserved when net external torque is zero.
Rotational equations mirror v = u + at etc. with ω, α, θ.

Remember: τ = Iα | L = Iω conserved | I of ring, disc, rod, sphere

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