# Nonuniform Circular Motion-Newtonian bucket problem

• Dougggggg
In summary, the problem involves swinging a pail of water in a vertical circle with a radius of 0.600 m and finding the minimum speed needed at the highest point to prevent the water from spilling. The solution can be found using the equation arad = v^2/R = 4π^2R/T^2, but it is easier to use conservation of energy. The correct answer is v = 2.43 m/s, which can be found by setting the potential energy at the highest point equal to the kinetic energy and solving for v. It is important to consider the tension in the cord to prevent it from going slack at the highest point.
Dougggggg

## Homework Statement

Word for word from textbook. . .
"You tie a cord to a pail of water, and you swing the pail in a vertical circle of a radius of 0.600 m. What minimum speed must you give the pail at the highest point of the circle if no water is to spill from it?"

R=0.600 m
g=9.80 m/s2

## Homework Equations

I tried to use this though I am unsure if it can be used since I believe this is a case of nonuniform circular motion.
arad = $$\frac{v^2}{R}$$ = $$\frac{4\pi^2 R}{T^2}$$

I have also tried using $$\Sigma$$F equations but those lead me into a mess.

## The Attempt at a Solution

I first tried to figure out what forces were involved at the top and bottom of the circle. I kept getting things that I couldn't really do anything with since I only knew one acceleration. I then went on to look at what I could do with arad and kept making a mess of algebra that wasn't really helping anything at all. I know how this problem works conceptually to some degree, the inertia makes the water resistant to change directions and so on, but I am not getting how I am supposed to do the math here. If someone could give me some idea of which direction I should start going I would be greatly appreciative.Edit: Found right answer, if I ignored tension as a force acting on the bucket then I came up with arad = g
then
$$\sqrt{gR}$$ = v

Solved for v. I am not sure how that gives me the minimum speed required to complete the circle but it said it was the right answer in the back of the book. If anyone could please explain why that math worked the way it did I would be really really really thankful.

Answer was v = 2.43 m/s

Edit 2: Does it work because for that whole equation to work v has to be big enough to complete a whole circle?

Last edited:
The usual centripetal motion way would require a tension or a normal reaction to give the speed 'v'.

It is easier this time to just use conservation of energy. If you consider a line through the center of the circle as your 0 energy line, at the highest point, the height is the radius 'R'.
It has associated PE 'mgR'.

At the highest point the bucket with water is moving at a speed 'v' and the associated KE is 0.5mv2.

You however just can't ignore the tension,as that would mean at the highest point, there is not tension at all, meaning that the cord would go slack and the bucket would fall to the ground shortly after reaching the highest point.

rock.freak667 said:
The usual centripetal motion way would require a tension or a normal reaction to give the speed 'v'.

It is easier this time to just use conservation of energy. If you consider a line through the center of the circle as your 0 energy line, at the highest point, the height is the radius 'R'.
It has associated PE 'mgR'.

At the highest point the bucket with water is moving at a speed 'v' and the associated KE is 0.5mv2.

You however just can't ignore the tension,as that would mean at the highest point, there is not tension at all, meaning that the cord would go slack and the bucket would fall to the ground shortly after reaching the highest point.

We haven't discussed circular motion to that extent yet. It probably can be noted as a force on the ex, it may just be tangent to the top of the circle. Therefore, with gravity, would cause an acceleration ahead of the normal. Which makes sense because it has the most acceleration at the top of the circle.

## 1. What is the Newtonian bucket problem?

The Newtonian bucket problem is a thought experiment proposed by Sir Isaac Newton in which a bucket of water is suspended by a rope and spun in a circular motion. The question is, what happens to the surface of the water as the bucket spins?

## 2. Why is it called "nonuniform" circular motion?

The motion of the bucket and water is considered nonuniform because the speed and direction of the water particles are constantly changing as the bucket spins. This is in contrast to uniform circular motion, where the speed and direction remain constant.

## 3. What causes the water to move to the edges of the bucket?

The water moves to the edges of the bucket due to centripetal force. As the bucket spins, the water particles experience a centrifugal force that pushes them outwards. However, the tension in the rope provides an equal and opposite force, causing the water to move towards the center and ultimately form a concave surface.

## 4. Does the water continue to spin after the bucket stops?

No, the water will come to rest once the bucket stops spinning. This is because there is no longer a centripetal force acting on the water particles and they will continue in a straight line until they are stopped by the sides of the bucket.

## 5. How does this thought experiment relate to Newton's laws of motion?

The Newtonian bucket problem demonstrates Newton's first law of motion, also known as the law of inertia. The water's natural tendency is to continue moving in a straight line, but the tension in the rope forces it to move in a circular path. This shows that an object will continue in its state of motion unless acted upon by an external force.

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