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Key Equations

Angular position θ=srθ=sr
Angular velocity ω=limΔt0ΔθΔt=dθdtω=limΔt0ΔθΔt=dθdt
Tangential speed vt=rωvt=rω
Angular acceleration α=limΔt0ΔωΔt=dωdt=d2θdt2α=limΔt0ΔωΔt=dωdt=d2θdt2
Tangential acceleration at=rαat=rα
Average angular velocity ω=ω0+ωf2ω=ω0+ωf2
Angular displacement θf=θ0+ωtθf=θ0+ωt
Angular velocity from constant angular acceleration ωf=ω0+αtωf=ω0+αt
Angular velocity from displacement and
constant angular acceleration
Change in angular velocity ωf2=ω02+2α(Δθ)ωf2=ω02+2α(Δθ)
Total acceleration a=ac+ata=ac+at
Rotational kinetic energy K=12(jmjrj2)ω2K=12(jmjrj2)ω2
Moment of inertia I=jmjrj2I=jmjrj2
Rotational kinetic energy in terms of the
moment of inertia of a rigid body
Moment of inertia of a continuous object I=r2dmI=r2dm
Parallel-axis theorem Iparallel-axis=Icenter of mass+md2Iparallel-axis=Icenter of mass+md2
Moment of inertia of a compound object Itotal=iIiItotal=iIi
Torque vector τ=r×Fτ=r×F
Magnitude of torque |τ|=rF|τ|=rF
Total torque τnet=i|τi|τnet=i|τi|
Newton’s second law for rotation iτi=Iαiτi=Iα
Incremental work done by a torque dW=(iτi)dθdW=(iτi)dθ
Work-energy theorem WAB=KBKAWAB=KBKA
Rotational work done by net force WAB=θAθB(iτi)dθWAB=θAθB(iτi)dθ
Rotational power P=τωP=τω
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