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When you have a mass moving on a rope in a vertical circular motion, at the top and the bottom of the swing, the force of tension cleanly takes care of gravity and creates a centripetal force. However, when the mass in the middle of a cycle like so: enter image description here

How does the force of gravity get counteracted? The force of tension is the only force that can provide the centripetal force since the only other force, gravity, is acting perpendicular to the radius. So, what force takes care of gravity? If gravity isn't somehow counteracted, then the net force would be the vector addition of the gravity and the force of tension which would be somewhere in between the two vectors and would not suffice as a centripetal force since it needs to be parallel to the radius. There is no way I can see a normal force of some sort to exist since the mass is not pushing off any object. So, how is the force of gravity taken care of? Thanks to anyone who can help.

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2 Answers 2

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System: point mass, $m$
External forces: gravitational attraction $\vec F_{\rm g} = F_{\rm g,r}\,\hat r + F_{\rm g,\theta}\,\hat \theta$ and tension in string $\vec F_{\rm T} = -F_{\rm T}\,\hat r$.

enter image description here

As long as the string is in tension, $\vec F_{\rm g} + \vec F_{\rm T} = m\,\vec a$ which can be divided into a radial component and a tangential component as shown in the right-hand diagram.

Radial: $-F_{\rm T} + F_{\rm g,r} = m \,a_{\rm centripetal}$
Tangential: $F_{\rm g,\theta} = m\,a_{\rm tangential}$

Note that the tension force is always radially inwards and always contributes to the centripetal acceleration but never affects the tangential acceleration.

Except at positions $A$ and $C$, the gravitational force produces a tangential acceleration and except at positions $B$ and $D$ the gravitational force has a contribution the centripetal acceleration.

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  • $\begingroup$ So, does the mass slow down towards the top of the swing because of the tangential component vector of gravity? $\endgroup$
    – Matthew
    Nov 10, 2022 at 9:15
  • $\begingroup$ Yes, and if this results in the mass moving too slowly the circular motion cannot be maintained and the string ceases to be taut. $\endgroup$
    – Farcher
    Nov 10, 2022 at 10:24
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The force of gravity is not counteracted.

It acts on the body with a constant force. This force is divided into tangential and radial components as the body moves to different positions around the vertical path.

The radial component, combined with the tension in the string, supplies the necessary centripetal force to keep the body moving in a circle.

The tangential component serves to give a tangential acceleration to the body. That's why the body slows down on the way up, and speeds up on the way down...

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  • $\begingroup$ So, in this case, the centripetal force (or net force) is not pointing toward the centre because the tension is also being added to the force of gravity which are perpendicular to each other? Then, how does the mass form a perfect circle if the net force does not point to the centre? Isn't a constant net force toward the centre non-negotiable for circular motion? $\endgroup$
    – Matthew
    Nov 10, 2022 at 7:45
  • $\begingroup$ The object is not moving a constant tangential velocity, so constant centripetal force is not required. $\endgroup$
    – DJohnM
    Nov 10, 2022 at 7:59

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