Top and bottom of a swinging object

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In a vertical circle with non-uniform speed, the tension in the swinging object is maximum at the bottom due to the need to provide centripetal force while also countering gravitational force. At this point, tension acts upward while gravity acts downward, leading to a higher net force requirement. Conversely, at the top of the swing, tension is at its minimum because gravitational force can assist in providing the necessary centripetal force, potentially leading to a scenario where tension could be zero if speed is just right. The discussion highlights the relationship between tension, gravitational force, and centripetal force in different positions of the swing. Understanding these dynamics is crucial for analyzing motion in circular paths.
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Homework Statement



In a vertical circle with non uniform speed, why we always say that the tension at the bottom is a maximum and the tension at the top is minimum?
Is there any mathematical method to prove it? I don't understand why we always assume in this way.


The Attempt at a Solution



could it because of energy?
 
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At the bottom the tension provides the centripetal force as usual but also must balance the gravitational force which is at that point entirely in the opposite direction to the tension force.
 
so could we say the tension is zero since both of the gravitational force and tension are in the same direction?
 
At the bottom of the swing tension is up, gravity down.
At the top, it could happen that they cancel out of the speed is just right.
 

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