Write τ = Iα, Then Convert Between Angular and Linear Quantities
Rotational dynamics is Newton's second law for rotation: net torque equals moment of inertia times angular acceleration. The rest is converting between revolutions, radians and linear quantities.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
τ = Iα and unit conversions
A solid cylinder of mass 50 kg and radius 0.5 m has I = 6.25 kg m², so an angular acceleration of 2 rev/s² (4π rad/s²) needs a tangential string tension T = I α / R = 6.25 x 4π / 0.5 = 50π, about 157 N.
Convert rpm or rev/s into rad/s before using any formula.
Linear and angular acceleration together
A point on the rim has a tangential acceleration aₜ = R α and a radial (centripetal) acceleration a_c = ω² R, which are perpendicular, so the total is sqrt(aₜ² + a_c²). A disc of radius 0.5 m starting from rest with α = 2 rad/s² has after 2 s: ω = 4 rad/s, aₜ = 1 m/s², a_c = 16 x 0.5 = 8 m/s², so the net acceleration is sqrt(65), about 8 m/s².
Which body accelerates faster?
For the same torque, the body with the smaller moment of inertia gets the larger angular acceleration. A metre scale made of a steel half and a wooden half pivoted at the steel end has less moment of inertia (the denser steel is close to the pivot) than when pivoted at the wooden end, so the same force at the free end gives more angular acceleration when the steel end is pivoted.
OlympIQ exam lens
- ✓τ = I α; T R = I α for a string on a cylinder
- ✓Convert rev/s to rad/s with a factor 2π before using τ = I α
- ✓Rim acceleration: tangential R α, radial ω² R, total sqrt of the sum of squares
- ✓Same torque: smaller I means larger α
Do not use the tangential acceleration alone as the net acceleration of a rotating point. Once the disc is moving the centripetal term ω² R is usually much larger and must be included with the tangential term.
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