University Physics Volume 1

# Key Equations

### Key Equations

 Relationship between frequency and period $f=1Tf=1T$ $Position in SHM withϕ=0.00Position in SHM withϕ=0.00$ $x(t)=Acos(ωt)x(t)=Acos(ωt)$ General position in SHM $x(t)=Acos(ωt+ϕ)x(t)=Acos(ωt+ϕ)$ General velocity in SHM $v(t)=−Aωsin(ωt+ϕ)v(t)=−Aωsin(ωt+ϕ)$ General acceleration in SHM $a(t)=−Aω2cos(ωt+ϕ)a(t)=−Aω2cos(ωt+ϕ)$ Maximum displacement (amplitude) of SHM $xmax=Axmax=A$ Maximum velocity of SHM $|vmax|=Aω|vmax|=Aω$ Maximum acceleration of SHM $|amax|=Aω2|amax|=Aω2$ Angular frequency of a mass-spring system in SHM $ω=kmω=km$ Period of a mass-spring system in SHM $T=2πmkT=2πmk$ Frequency of a mass-spring system in SHM $f=12πkmf=12πkm$ Energy in a mass-spring system in SHM $ETotal=12kx2+12mv2=12kA2ETotal=12kx2+12mv2=12kA2$ The velocity of the mass in a spring-masssystem in SHM $v=±km(A2−x2)v=±km(A2−x2)$ The x-component of the radius of a rotating disk $x(t)=Acos(ωt+ϕ)x(t)=Acos(ωt+ϕ)$ The x-component of the velocity of the edge of a rotating disk $v(t)=−vmaxsin(ωt+ϕ)v(t)=−vmaxsin(ωt+ϕ)$ The x-component of the acceleration of theedge of a rotating disk $a(t)=−amaxcos(ωt+ϕ)a(t)=−amaxcos(ωt+ϕ)$ Force equation for a simple pendulum $d2θdt2=−gLθd2θdt2=−gLθ$ Angular frequency for a simple pendulum $ω=gLω=gL$ Period of a simple pendulum $T=2πLgT=2πLg$ Angular frequency of a physical pendulum $ω=mgLIω=mgLI$ Period of a physical pendulum $T=2πImgLT=2πImgL$ Period of a torsional pendulum $T=2πIκT=2πIκ$ Newton’s second law for harmonic motion $md2xdt2+bdxdt+kx=0md2xdt2+bdxdt+kx=0$ Solution for underdamped harmonic motion $x(t)=A0e−b2mtcos(ωt+ϕ)x(t)=A0e−b2mtcos(ωt+ϕ)$ Natural angular frequency of amass-spring system $ω0=kmω0=km$ Angular frequency of underdampedharmonic motion $ω=ω02−(b2m)2ω=ω02−(b2m)2$ Newton’s second law for forced,damped oscillation $−kx−bdxdt+Fosin(ωt)=md2xdt2−kx−bdxdt+Fosin(ωt)=md2xdt2$ Solution to Newton’s second law for forced,damped oscillations $x(t)=Acos(ωt+ϕ)x(t)=Acos(ωt+ϕ)$ Amplitude of system undergoing forced,damped oscillations $A=Fom2(ω2−ωo2)2+b2ω2A=Fom2(ω2−ωo2)2+b2ω2$