Interactive Lecture Demonstration 9 – Bucket of Water

A student will whirl a bucket of water in a vertical circle. The student will slowly decrease the speed of the bucket, until the water just barely stays in contact with the bucket. You will measure the period of revolution of the bucket of water for that situation and compare with the period you predict based on a force diagram analysis.

1. First, the prediction. We want to know the maximum period of the bucket so that the water will stay in contact with the bucket. It should be obvious to you that the water is most likely to fall out of the bucket when the bucket is upside-down, thus we need to focus on the part of the motion when the bucket and water are at the very top of their vertical circle. Draw below a diagram of the bucket and water for that situation. Next to your diagram, draw the force diagram for the water. We need the water's force diagram, not the bucket's, since the water is the object whose motion we really care about in this problem. Common sense should tell you what forces to draw on the force diagram, and what direction those forces must have.  

2. Follow the usual procedure now to choose the positive direction and write the net force equation for the water.

3. Now we need to find a way to solve for the maximum period of the bucket so that the water will stay in contact with the bucket. We hope you already realized that this is an inequality problem, since we are looking for a maximum value of something. As with any inequality problem, we must write down an inequality that governs the situation. Later we will substitute from our net force equation into the inequality and get a formula for the quantity we want to know (in this case, the period P). To find the correct inequality that governs this problem, read carefully again the problem statement in bold italics above. Write down an inequality (involving an appropriate physics variable) that says "the water will stay in contact with the bucket." 
4. We hope you realized that as long as the water is in contact with the bucket, there is a normal force n of the bucket on the water. If the water loses contact with the bucket, n vanishes. Thus the condition for the water to stay in contact with the bucket can be written as n > 0. Note that n can never be less than 0, since that would mean the direction of n is opposite to the direction we drew it on our force diagram, and if n were opposite to the direction we drew it then the bucket would have to be sucking the water in (instead of pushing on the water, as all real buckets do).

Now that you have the inequality that governs this problem, you can substitute into the inequality an expression for n that you get from the net force equation. Note that you may NOT substitute into the net force equation the value zero for n, since n does NOT equal zero ! (n is greater than or equal to zero). 

Make your substitution now, then make another substitution to express the centripetal acceleration in terms of the period and radius of the water's motion. Finally, rearrange your formula to get an inequality that shows the maximum value of the period in terms of the variables R and g only (plus some constant factors).

5. Now we need to make some actual measurements. We need to measure the radius of the water's circle. Between what points should we measure the distance in order to find that radius ? After measuring the radius, calculate the maximum predicted period according to the formula you derived in #4.

6. Now someone will whirl the bucket. When you hear the water sloshing so that it is almost going to fall out of the bucket, start timing. Time at least three consecutive revolutions in order to get a good average measurement of the period. Record your measurement of the period below and compare it with the value you predicted.