← All printables Chapter 08 · Mechanics · 2 sheets · read the chapter →
Vedatom Physics
FORMULA CARD · PHYSICS.VEDATOM.COM
Chapter 08 · Part I · Mechanics
Revised on ____________________
Chapter 08 · Rotational Mechanics

Every linear law, once more with r in it

2 sheets
2 diagrams
29 results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

I

Rotational kinematics

SAME FIVE EQUATIONS, NEW SYMBOLS
I.Angular velocity  ω = dθdt , α = dωdt
II.Constant α  ω = ω0 + α t
III.  θ = ω0 t + 12 α t²
IV.  ω² = ω0² + 2 α θ
V.Bridge to linear  v = r ω , at = r α
VI.Rigidity  every particle shares ω and α not v, not a
II

Torque and moment of inertia

I DEPENDS ON THE AXIS, ALWAYS
I.Torque  τ = r × F , τ = r F sin θ
II.Lever arm  τ = F × (⊥ distance to the line of action)
III.Newton's law, rotational  Στ = I α
IV.Moment of inertia  I = Σ mi ri² = ∫ r² dm
V.Parallel axis  I = Icm + M d²
VI.Perpendicular axis  Iz = Ix + Iy plane laminae only
VII.Radius of gyration  k = √( IM )
VIII.Equilibrium  ΣF = 0 and Στ = 0 torques about any point
RingM R2Disc½ M R2Solid sphere⅗ M R2Rod · centre1/12 M L2Rod · end⅓ M L2Hollow sphere⅔ M R2
III

Angular momentum

CONSERVED WHEN THE NET TORQUE IS ZERO
I.A particle  L = r × p , L = m v r sin θ
II.A rigid body  L = I ω
III.The law  τext = dLdt
IV.Conservation  I1 ω1 = I2 ω2 the skater pulling her arms in
V.Energy then  K = L²2 I rises as I falls — she does the work
VI.Precession  Ω = τL = M g rI ω
VII.About the cm  L = Lcm + Labout cm orbital + spin
IV

Rolling without slipping

THE CONSTRAINT DOES THE WORK
I.Constraint  vcm = R ω , acm = R α
II.Down an incline  a = g sin θ1 + k²/R²
III.No slipping if  μ ≥ tan θ1 + R²/k²
IV.The rolling race  smallest k²/R² wins sphere ⅖ < disc ½ < ring 1
v2v0√2 v
V

Rotational work and energy

THE LAST COLUMN
I.Work  W = ∫ τ dθ
II.Power  P = τ ω
III.Kinetic energy  K = 12 I ω²
IV.Combined motion  K = 12 M vcm² + 12 Icm ω²
Where marks are lost in this chapter
1Writing two correct equations about two different axes. Choose the axis first, then keep it.
2Quoting I without naming the axis. MR²/2 is a disc about its centre and nowhere else.
3Using the perpendicular-axis theorem on a solid body. It holds for plane laminae only.
4Letting rolling friction dissipate energy. The contact point does not move, so its work is zero.