Inverted Pendulum: Relating Force to Angle/Displacement

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The discussion focuses on analyzing the dynamics of an inverted pendulum positioned at an angle theta from the horizontal, with a force applied to its end that decreases as the angle increases. The critical point occurs when the pendulum passes the vertical, leading to a fall on the opposite side. Participants suggest that the weight at the end generates a torque calculated as mgr*cos(theta), where r is the pendulum's length. To counteract this torque, a minimum force of mg*cos(theta) is required if applied perpendicularly to the rod. Understanding these relationships is essential for solving the problem effectively.
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Im considering a problem were an inverted pendulum is resting against a ledge an an angle theta from the horizontal. A force is applied to the end causing it to move, This force is dynamic and decrease as the angle increases. The critical point is when the pendulum passes the vertical and cannot return to its resting point falls to the other side.

Does anyone know how i can relate the magnitude of the force to either the angle or displacement of the end of the pendulum?

suggestions welcome.
thanks
 
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The weight at the end of the pendulum produces a torque. This is obviously mgr*cos(theta), where r is the length of the pendulum. To counter this torque, the minimum force needed is mg*cos(theta) if the force is directed perpendicularly to the rod.
 
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