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In vertical circular motion we conserve energy for calculating velocities at a point (if initial velocity given). But, energy can only be conserved when forces are conservative. Tension is not a conservative force. Does it not affect the particles velocity? Is the tension's role only to provide centripetal acceleration?

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  • $\begingroup$ What makes you say that tension isn't conservative? $\endgroup$ – Will Cross Nov 23 '13 at 22:52
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By definition, the tension in the string can only supply a force towards the center of the circular motion, so in one sense the tension is a centripetal force.

However, at any moment, the force of gravity can be decomposed into radial and tangential components. It is the sum of the tension and the inward radial component of gravity that supplies the necessary centripetal force.

For example, with the right starting conditions, the velocity of the mass as it "goes over the top" will be $\sqrt{Rg}$, and gravity will supply all the centripetal force needed; at that moment, the tension would be zero.

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Yes, tension only affects the direction of the particle's velocity. This is because it is always perpendicular to the velocity, and because work is actually the dot product of force and displacement:

$$ W = F \cdot s = |F| \times |s| \times \cos(\theta) $$

, a force perpendicular to the displacement does no work

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