Astudent has arms of length 63 cm. what is the minimum angular velocity (in rad/s) at which a bucket of water can be swing to prevent spilling water? the distance from the handle to the standard water level in the bucket is 20 cm.

For the answer to the question above,  The condition for a vertical circle is that the centrifugal acceleration has to exceed that of gravity. 

v^2 / r = omega^2 r > g 

So the minimum angular speed is: 
omega > sqrt (g/r) 

You know g. r is a bit tricky and I don't like what they've given you. The r you are worried about is from the axis of rotation (shoulder) to the top of the water in the bucket--not the bottom because that is the water which experiences the least centrifugal force. We'll assume that there isn't much water in there, so just use the arm length plus the handle-to-bottom length for the radius that you plug in.