University Physics Volume 1

# Key Equations

### Key Equations

 Angular position $θ=srθ=sr$ Angular velocity $ω=limΔt→0ΔθΔt=dθdtω=limΔt→0ΔθΔt=dθdt$ Tangential speed $vt=rωvt=rω$ Angular acceleration $α=limΔt→0ΔωΔt=dωdt=d2θdt2α=limΔt→0ΔωΔt=dωdt=d2θdt2$ Tangential acceleration $at=rαat=rα$ Average angular velocity $ω–=ω0+ωf2ω–=ω0+ωf2$ Angular displacement $θf=θ0+ω–tθf=θ0+ω–t$ Angular velocity from constant angular acceleration $ωf=ω0+αtωf=ω0+αt$ Angular velocity from displacement andconstant angular acceleration $θf=θ0+ω0t+12αt2θf=θ0+ω0t+12αt2$ Change in angular velocity $ωf2=ω02+2α(Δθ)ωf2=ω02+2α(Δθ)$ Total acceleration $a→=a→c+a→ta→=a→c+a→t$ Rotational kinetic energy $K=12(∑jmjrj2)ω2K=12(∑jmjrj2)ω2$ Moment of inertia $I=∑jmjrj2I=∑jmjrj2$ Rotational kinetic energy in terms of themoment of inertia of a rigid body $K=12Iω2K=12Iω2$ 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 $τ→=r→×F→τ→=r→×F→$ Magnitude of torque $|τ→|=r⊥F|τ→|=r⊥F$ Total torque $τ→net=∑i|τ→i|τ→net=∑i|τ→i|$ Newton’s second law for rotation $∑iτi=Iα∑iτi=Iα$ Incremental work done by a torque $dW=(∑iτi)dθdW=(∑iτi)dθ$ Work-energy theorem $WAB=KB−KAWAB=KB−KA$ Rotational work done by net force $WAB=∫θAθB(∑iτi)dθWAB=∫θAθB(∑iτi)dθ$ Rotational power $P=τωP=τω$