What Is the Minimum Speed to Prevent Water Spillage in a Rotating Pail?

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To prevent water spillage from a rotating pail at the top of a vertical circle with a radius of 1 meter, the minimum speed must ensure that the centripetal force is sufficient to counteract gravity. The equation derived, t + mg = m(v^2/r), indicates that the tension (t) plus the weight (mg) must equal the centripetal force (m(v^2/r)). As long as the tension remains non-zero, the water will stay in the pail; if it reaches zero, spillage occurs. The discussion highlights the importance of maintaining a minimum speed to create necessary centripetal force. Understanding these dynamics is crucial for solving problems related to nonuniform circular motion.
lunarskull
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nonuniform circular motion

a pail of water is rotated in a vertical circle of radius 1m. what is the minimum speed of the pail at the top of the circle if no water is to spill out?

hmmm this problem is difficult for me to do because they gave only 1 variable. i drew a freebody diagram (i doubt if i did it right) and came up with the equation t+mg=m(v^2/r) am i doing this right?
 
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Right, assuming that "t" is the force exerted by the pail on the water. As long as the pail exerts some non-zero force, the water remains in the pail; when that force goes to zero, the water begins to spill out.
 

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