Acollege student holds a pail full of water by the handle and whirls it around in a vertical circle at a constant speed. the radius of this circle is 0.95 m. what is the minimum speed that the pail must have at the top of its circular motion if the water is not to spill out of the upside-down pail

Here we have a circular motion. In that type of motion two new forces appear: centrifugal and centripetal. Centrifugal force tries to move object away frome the center of rotation. Centripetal force is the reason why object moves in a circle.

According to Newton's first law, in order for water not to spill out total force acting on water must be equal to zero. Forces acting on the water are gravitational and centrifugal.
" alt="F_{g} = F_{c} \\ m*g= \frac{m* v^{2} }{r} \\ g= \frac{ v^{2} }{r} \\ v^{2} =g*r \\ v= \sqrt{g*r} \\ v= \sqrt{9.81*0.95} \\ v= \sqrt{9.3195} \\ v=3.1 m/s" /> " />

Forces have opposite signs as they act in opposite directions. But their value is the same so we can ignore signs:

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