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In the picture above, why is it only necessary that the net force in the radial direction is equal to $\frac{mv^2}{r}$ for the object to complete circular motion? What about the component of gravity that acts perpendicular to $F_{tension}$?

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  • $\begingroup$ That component does not cause the circular motion. It only causes some speeding up. So if you are working on the circular motion, then it is irrelevant $\endgroup$ – Steeven Oct 6 '17 at 8:02
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The force perpendicular to the $F_{Tension}$ is used in decelerating the speed of the ball. Note, however that the same couldn't be applied to centripetal force as the particle isn't accelerating radially throughout the motion. So, the forces must balance out.

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