Circular Motion/Newton's Laws-- Swinging a Pail

[in the rye]

Homework Statement

You tie a cord to a pail of water, and you swing the pail in a vertical circle of radius 0.700 m
What minimum speed must you give the pail at the highest point of the circle if no water is to spill from it?

Homework Equations

The Attempt at a Solution

I got the correct answer. 2.62 m/s, but I'm not entirely sure WHY I got the correct answer. So, what I did was I drew a force diagram and labeled all my forces. I got that my my forces in the centripetal direction are Tension (F_T) and Gravity (mg). I found no forces in tangential direction.

Setting up my Newtons 2nd Law equation I got:

F_T + m g = m a
or
F_T + m g = m (v²/r)

When I looked at this I basically used some intuition. I figured since m,g, and r are considered to be constants, the only thing I could do is eliminate my tension force by setting it to 0. This made sense to me to a degree because if I set tension to 0 then only gravity is acting on my water, which meant the right side (ma) would be equal to gravity (mg) to keep the water in place.

However, I'm not really sure why this means I would set tension to 0. I basically tried to imagine if I had complete slack in the chord what would happen, and intuitively my answer made sense. I'm just not sure of the math behind WHY it makes sense. It was more or less a lucky guess with a bit of intuitive inclination on my part. Why did this work out?

 

[Biker]

Notice what the question said "Minimum"
Imagine having a tension force all right? You will need a bigger velocity to balance things out.
Imagine now decreasing the tension force, as a result the velocity will decrease.
Now set it to 0. The velocity will be much smaller than what it was.
Can you decrease the force of gravity?
So there you have the minimum speed.

 

[in the rye]

Notice what the question said "Minimum"
Imagine having a tension force all right? You will need a bigger velocity to balance things out.
Imagine now decreasing the tension force, as a result the velocity will decrease.
Now set it to 0. The velocity will be much smaller than what it was.
Can you decrease the force of gravity?
So there you have the minimum speed.

I kind of see where you're getting. That is, at 0 the chord goes slack. But, how can I ensure the water will stay in the bucket? I guess that's what I'm confused with. If my only force acting on it is the acceleration due to gravity, I'd imagine that the water would fall out (which makes it seem like there's a centrifugal force -- even though I know there isn't).

[edit]

Rethinking the problem, this reminds me of projectile motion a little. If my tension is 0, then my bucket is kind in free fall, keeping the water and everything inside it since EVERYTHING is falling, however, since the velocity is tangential to the circle, it'd have a horizontal velocity giving it a parabolic type motion. Right?

However, when I have a higher tension, the water is staying put because of Newton's first law, correct? That is, it wants to launch horizontally, but the normal force of the bucket on the water is keeping it in place.

 

[haruspex]

Yes, and the free fall parabola is tangential to the circular motion at the top, i.e. has the same radius of curvature there.

However, when I have a higher tension, the water is staying put because of Newton's first law, correct? That is, it wants to launch horizontally, but the normal force of the bucket on the water is keeping it in place.

Nearly right. Which way is that normal force acting?

 

[in the rye]

Yes, and the free fall parabola is tangential to the circular motion at the top, i.e. has the same radius of curvature there.

Nearly right. Which way is that normal force acting?

I guess it'd depend what part we're looking at but I'd imagine it'd be acting downwards at its peak, upwards at its valley, in fact, it seems like the normal force would be centripetal as well assuming the opening of the bucket is always faced towards the center?

 

[haruspex]

I guess it'd depend what part we're looking at but I'd imagine it'd be acting downwards at its peak, upwards at its valley, in fact, it seems like the normal force would be centripetal as well assuming the opening of the bucket is always faced towards the center?

Yes, but you don't need that assumption. The normal force plays the same role for the water that the string plays for the bucket+water. The two forces would be in the ratio of the masses.
My quibble was because you wrote

it wants to launch horizontally