What is the time period of a pendulum on an inclined plane?

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The discussion focuses on determining the time period of a pendulum suspended from a cart sliding on an inclined plane. The effective gravitational acceleration (geff) must be calculated, as the pendulum experiences gravity differently due to the incline. The equation T=2(pi)√(l/geff) is relevant, but the challenge lies in accurately finding geff in this scenario. Participants express confusion about deriving the time period for pendulums on inclines compared to standard pendulum cases. Understanding the direction of geff and its application is crucial for solving the problem effectively.
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Period of a pendulum...

Homework Statement


A simple pendulum of length l is suspended from the ceiling of a cart which is sliding without friction on an inclined plane of inclination theta. What will be the time period of the pendulum?


Homework Equations


T=2(pi) root(l/geff)



The Attempt at a Solution


There is this concept of geff(g effective) which i don't know how to apply...So basically how to do this sum?
 
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when the (pendulum + cart)system moves along the inclined plane, they do not feel the effect of any gravity in that direction as if they were in free fall in the sliding direction.
so, what remains is the g component perpendiculur to the slide direction, which is your effective g in this case.
 


Ya...that is okay...for this sum...but in general how to find the direction of g(eff)...
I mean for some general cases there must be some way to find the time period using the general method ...ie showing that the a=-w^2*x or torque=-w^2(theta) for the system and then using T=2(pi)/w... For other cases i have done it many times...even for a normal pendulum(the commom derivation)...But i am not getting such a method for tilted pendulums...
 

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Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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