Circular Motion and Tension

[ieatshrimp24]

When you swing a ball on a string in a VERTICAL motion and want to measure the tension, the equation looks like Fc = T + mg = mv^2/r where Fc is the centripetal force.
When you swing a ball on a string in a HORIZONTAL motion and want to measure the tension, the equation looks like Fc = T = mv^2/r.

Can anyone explain the differences in these two equations?

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[Odi]

When you swing a ball in the vertical plane the tension will be in the same direction as the balls weight at the apex of the spin and in the opposite direction at the bottom of the spin.

This results in two different Fc formulas... apex is T + mg = ma and bottom is T - mg = ma. See pic:

When you spin the ball in a perfectly horizontal plane the tension will never point in the same plane as the balls weight. The balls weight will point down in the y-direction having no effect on the centripetal force which is now always in the x-direction pointing towards the center of the spin. This results in one formula for Fc: T = ma.

 

[azor ahai]

In a horizontal plane, you only account for the centripetal force, so you can set that equal to the tension in the string.

In a vertical plane, you account for both the centripetal force and the force due to gravity. At the apex, like the poster above stated, has them both in the same direction so you add them. At the bottom, they're in opposite directions so you subtract them.