C10 - Rotation of Rigid Objects Abt Fixed Axis

Angular position

• s = rθ

• θ_rad = (π/180)θ_dgr

Kinematic eqns (rotational)

Angular & translational qtys

• v = rω

• a = rα

• a_c = rω²

• a_total = r(sqrt(α²+ω⁴))

Torque

• T = rFsinΦ

• ΣT = Iα

Moment of inertia

• Extent to which an object is willing to change in rotational speed, based on mass distribution

• Particle mass: I = mr²

  • Differs based on object type

Hoop

I = MR²

Solid cylinder/disk

I = MR²/2

Hollow cylinder

I = M(R₁²+R₂²)/2

Rectangular plate

I = M(l²+w²)/12

Thin rod

• I = ML²/12 (center axis)

• I = ML²/3 (end axis)

Sphere

• I_solid = 2MR²/5

• I_shell = 2MR²/3

Parallel-axis thm

I_total = I_CM + MD²

• D = axis → CM

Rotational KE

• K_r = Iω²/2

  • Same value as K_{translational}

Rotational power

P = τω

Rotational work thm

W = I(ω_f²-ω_i²)/2