How can Simple Harmonic Motion have angular frequency?

jtbell

See the section of this video from about 1:45 to 2:07. It demonstrates the connection between SHM and uniform circular motion.

http://www.youtube.com/watch?v=SZ541Luq4nE

InvalidID

Derived it!

[tex]TE={ KE }+PE\quad and\quad TE={ KE }_{ max }={ PE }_{ max }\\ \\ { KE }_{ max }=KE+PE\\ \frac { 1 }{ 2 } m{ { v }_{ max } }^{ 2 }=\frac { 1 }{ 2 } m{ { v } }^{ 2 }+\frac { 1 }{ 2 } k{ x }^{ 2 }\\ m{ { v }_{ max } }^{ 2 }=m{ { v } }^{ 2 }+k{ x }^{ 2 }\\ k{ x }^{ 2 }=m{ { v }_{ max } }^{ 2 }-m{ { v } }^{ 2 }\\ { x }^{ 2 }=\frac { m }{ k } ({ { v }_{ max } }^{ 2 }-{ { v } }^{ 2 })\\ { x }^{ 2 }={ ω }^{ -2 }({ { v }_{ max } }^{ 2 }-{ { v } }^{ 2 })\\ x=\pm \sqrt { { ω }^{ -2 }({ { v }_{ max } }^{ 2 }-{ { v } }^{ 2 }) } \\ x=\pm \frac { \sqrt { { { v }_{ max } }^{ 2 }-{ { v } }^{ 2 } } }{ { ω } } [/tex]

WannabeNewton

Nice!

InvalidID

[tex]F(t)=F_{ 0 }sin(ωt)[/tex]

Is there a reason why the driving oscillator in forced simple harmonic motion doesn't have a phase constant?

sophiecentaur

It would only be an added complication.