Lagrangian for a rod pivoting at the end of a clock hand

[ChiralSuperfields]

Homework Statement

Please see belwo

Relevant Equations

##ml^2\ddot \phi + mRl\varphi \dot \phi\sin(\phi - \varphi t) = 0##

For (d),

I am confused how ##a(t)## oscillates harmonically. The EOM is ##ml^2\ddot \phi + mRl\varphi \dot \phi\sin(\phi - \varphi t) = 0##

Then using ##\phi - \varphi t = a(t)##
##l\ddot \phi + R\phi \varphi a(t) = 0##
Where I used the small angle approximation after substituting in ##a(t)##
Does anybody please know where to go from here to find ##\omega##?

Thanks!

 

[Orodruin]

Simply linearise the equation of motion around ##\phi^*##. You have attempted this but need to double check your math.

Also note: It is ##\gamma##, not ##\varphi##.

 

[ChiralSuperfields]

Simply linearise the equation of motion around ##\phi^*##. You have attempted this but need to double check your math.

Also note: It is ##\gamma##, not ##\varphi##.

Thank you for your reply @Orodruin!

##l^2\ddot \phi + Rl\phi \gamma a(t) = 0## is that please correct now?

Sorry I don't understand how you linearize the equation of motion around ##\phi^*##. I have not been taught that. Could you please explain a bit more?

Thanks!

 

[Orodruin]

Thank you for your reply @Orodruin!

##l^2\ddot \phi + Rl\phi \gamma a(t) = 0## is that please correct now?

No. Carefully consider the second term. Where did the ##\phi## come from?

Sorry I don't understand how you linearize the equation of motion around ##\phi^*##. I have not been taught that. Could you please explain a bit more?

It is essentially what you are (attempting) to do. You write ##\phi = \phi^* + a(t)## and then assume that ##a(t)## is small. Perform a Taylor expansion in ##a(t)## and keep only linear terms.

 

[Orodruin]

Also, for the first term, you have not inserted the assumption ##\phi = \phi^* + a## ...