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T Shaped Pendulum

https://isaacphysics.org/questions/t_pendulum_num?board=c50b0214-8984-4311-b93f-f7a553aaf506&stage=further_a

I managed to get the answer to this question but I dont really understand why?

I got the equation KE = 1/2 * I * θ'^2 + (3Lmg)/2 * (1- θ^2/2)
dE/dt = 0 so
θ" = (3Lmg)/(2I) θ
θ" = w^2 θ

so you can work out w and hence T.

My question is
usually rotational KE = 1/2 * I * w^2
so I was like ok they just used w = θ'
but then you have to use T = 2pi/w, which doesnt really make sense to me because w = θ' is a function of time, so how can there be a set time period?

Any explanation greatly appreciated (and I got I =17/12 *mL^2)
(edited 1 month ago)
Reply 1
Original post by mosaurlodon
https://isaacphysics.org/questions/t_pendulum_num?board=c50b0214-8984-4311-b93f-f7a553aaf506&stage=further_a
I managed to get the answer to this question but I dont really understand why?
I got the equation KE = 1/2 * I * θ'^2 + (3Lmg)/2 * (1- θ^2/2)
dE/dt = 0 so
θ" = (3Lmg)/(2I) θ
θ" = w^2 θ
so you can work out w and hence T.
My question is
usually rotational KE = 1/2 * I * w^2
so I was like ok they just used w = θ'
but then you have to use T = 2pi/w, which doesnt really make sense to me because w = θ' is a function of time, so how can there be a set time period?
Any explanation greatly appreciated (and I got I =17/12 *mL^2)

Without working it through,
θ' != w
for shm. Its true for (constant speed) circular motion when θ' is constant (angular acceleration is zero), so
θ = wt
but as you say for shm
θ' = -Awsin(wt+phi)
or something like that. Its a function of time and w is the angular frequency.

In the video, the w is angular velocity so θ', rather than the w (angular frequency) in the shm equation.
(edited 1 month ago)
ok I was a little confused because the equation for KE in hint 4 and 5 basically say θ' = w but I think I get it now.

so normally in circular motion θ' = w but in this case we're using shm so w = angular frequency not angular velocity.

rotational KE = 1/2 * I * (angular velocity)^2 = 1/2 * I * θ'^2
and T = 2*pi/(angular frequency) = 2*pi/w

Thank you for clearing that up

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