- #1
maverick280857
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Hi
Here is a problem I need some help with:
Find the time period of a simple pendulum (length l, mass m) when it is attached to the roof of a car moving in a circle with constant speed v and radius r.
Here is what I did:
IF the angle made by the string (inextensible) with the vertical at some instant is [itex]\theta[/itex] then the equations of motion in the direction along the length of the string and perpendicular to it are:
[tex]F_{normal} = mg\cos\theta - \frac{mv^2}{r}\sin\theta[/tex]
[tex]F_{tangential} = -ml \frac{d^2\theta}{dt^2} = mg\sin\theta - \frac{mv^2}{r}\cos\theta[/tex]
Using the small angle approximation in the second equation, we get
[tex]l\frac{d^2\theta}{dt^2} = -g\theta - \frac{mv^2}{r}[/tex]
This gives [itex]T = 2\pi\sqrt{\frac{l}{g}}[/itex] whereas the answer in my book is,
[tex]T = 2\pi\sqrt{\frac{l}{\sqrt{ g^2+(\frac{v^2}{r})^2 }}}[/tex]
I would be grateful if someone could help me out with this (I know v^2/r has got to play a role but how??) at the earliest...
Cheers
vivek
Here is a problem I need some help with:
Find the time period of a simple pendulum (length l, mass m) when it is attached to the roof of a car moving in a circle with constant speed v and radius r.
Here is what I did:
IF the angle made by the string (inextensible) with the vertical at some instant is [itex]\theta[/itex] then the equations of motion in the direction along the length of the string and perpendicular to it are:
[tex]F_{normal} = mg\cos\theta - \frac{mv^2}{r}\sin\theta[/tex]
[tex]F_{tangential} = -ml \frac{d^2\theta}{dt^2} = mg\sin\theta - \frac{mv^2}{r}\cos\theta[/tex]
Using the small angle approximation in the second equation, we get
[tex]l\frac{d^2\theta}{dt^2} = -g\theta - \frac{mv^2}{r}[/tex]
This gives [itex]T = 2\pi\sqrt{\frac{l}{g}}[/itex] whereas the answer in my book is,
[tex]T = 2\pi\sqrt{\frac{l}{\sqrt{ g^2+(\frac{v^2}{r})^2 }}}[/tex]
I would be grateful if someone could help me out with this (I know v^2/r has got to play a role but how??) at the earliest...
Cheers
vivek
Last edited: