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:
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.
Best Answer
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 $\vec F_{\rm T} = -F_{\rm T}\,\hat r$.
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.