Understanding Vertical Circular Motion: Why Do We Set Normal Force to Zero?

[madah12]

In a lot of physics book there is an example of a problem saying that we tie a bucket of water to a string and move it with vertical circular motion what is the minimum speed ,and the way they told us to solve these problem is to set the normal force to 0 then solve for v ,but I never actually got why is that? I mean when there is higher normal force wouldn't the water have a higher tendency to fall? because the bucket is pushing on it with greater force then it will have greater acceleration that is pointing downwards ,so can you tell me why do we set the normal force to zero and why is my understanding that the normal force will increase its tendency to fall is wrong?

 

[madah12]

also second thing I don't understand is: does the water have no vertical velocity at the top because velocity is perpendicular to the acceleration so it is to the right or left but then there shouldn't be be a normal force right? because the water isn't being pressed against the bucket.

 

[The_Duck]

There are two ways of looking at it.

Say you are an ant sitting on the rim of the bucket, which is being swing around in circles. You will feel two forces acting on you: gravity [mg], whose direction rotates around in a circle as the bucket rotates, and the centrifugal force [mv^2/r], which always points toward the bottom of the bucket. The water in the bucket also feels these forces. You expect the water to rise out of the bucket if the net force on it is upward, which could happen if the bucket inverted and the gravitational force is stronger than the centrifugal force. On the other hand if the centrifugal force is always stronger than gravity the water will always feel a net force pushing it down into the bucket, and will stay in. So you can compare the magnitude of the centrifugal force and gravity to see what will happen.

Now think about it from the perspective of someone on the ground. First imagine you had an inverted bucket full of water, and you let go of the bucket. Will the water fall out of the bucket? No, because under the influence of gravity both the water and the bucket accelerate downward at the same rate [g]. If you pull the bucket down faster than the rate at which it would naturally fall, the water will be pressed in and will be even less likely to come out. On the other hand, if you slow the bucket down as it's falling the water will keep going and come out.

If you are instead swinging the bucket around in circles, it always has an acceleration v^2/r toward the center of the circle. In particle, at the top of the circle, that acceleration is directed downward. Meanwhile gravity wants to accelerate the water downward at a rate g. As above, if the water wants to accelerate down faster than the bucket is accelerating down [g > v^2/r] the water will fall down out of the bucket. But if you are pulling the bucket down faster than gravity would naturally make it fall [v^2/r > g] then the water gets pressed into the bucket and doesn't come out.

At the critical speed, v^2/r = g, the acceleration of the bucket at the top of its arc is equal to gravitational acceleration, so you don't have to apply any force with your arm at that point, the bucket just sort of "floats" through the top of the arc. Perhaps you have experienced this.

Re: your question about velocity: An object's velocity points along the path it travels. So the bucket's velocity always points along the circle. For acceleration I like to think of two cases. If the acceleration is directed along the path of travel, it speeds up or slows down the object. If the acceleration is directed perpendicular to the path of travel, it changes the direction of motion. If you are swinging a bucket around a circle at a constant speed, the speed never changes, so the acceleration is wholly perpendicular to the direction of motion, with no parallel component.

 

[madah12]

There are two ways of looking at it.

Say you are an ant sitting on the rim of the bucket, which is being swing around in circles. You will feel two forces acting on you: gravity [mg], whose direction rotates around in a circle as the bucket rotates, and the centrifugal force [mv^2/r], which always points toward the bottom of the bucket. The water in the bucket also feels these forces. You expect the water to rise out of the bucket if the net force on it is upward, which could happen if the bucket inverted and the gravitational force is stronger than the centrifugal force. On the other hand if the centrifugal force is always stronger than gravity the water will always feel a net force pushing it down into the bucket, and will stay in. So you can compare the magnitude of the centrifugal force and gravity to see what will happen.


At the critical speed, v^2/r = g, the acceleration of the bucket at the top of its arc is equal to gravitational acceleration, so you don't have to apply any force with your arm at that point, the bucket just sort of "floats" through the top of the arc. Perhaps you have experienced this.

Re: your question about velocity: An object's velocity points along the path it travels. So the bucket's velocity always points along the circle. For acceleration I like to think of two cases. If the acceleration is directed along the path of travel, it speeds up or slows down the object. If the acceleration is directed perpendicular to the path of travel, it changes the direction of motion. If you are swinging a bucket around a circle at a constant speed, the speed never changes, so the acceleration is wholly perpendicular to the direction of motion, with no parallel component.

I got your second example but I didn't get the first one first what is the centrifugal force ,and second why would the net force be upward if the gravitational force is higher than that force?

also about the velocity part I was saying that since direction of the acceleration is toward the center at the top the velocity should be horizontal so both the water and the bucket shouldn't have a vertical velocity and they are falling at the same rate so how will they press against each other?

 

[The_Duck]

I got your second example but I didn't get the first one first what is the centrifugal force ,and second why would the net force be upward if the gravitational force is higher than that force?

You can Google centrifugal force. Say you are on a merry-go-around that is spinning very fast. You can imagine that there would be a "force" trying to throw you outward off the merry-go-round. It's the force you feel that pushes you to the side when you go around a tight curve in a car. That's the idea. I was considering the situation where you are an ant hanging on to the rim of the bucket as it is inverted. Then gravity points "upward" from your perspective, and will pull the water out of the bucket if centrifugal force doesn't counteract it.

also about the velocity part I was saying that since direction of the acceleration is toward the center at the top the velocity should be horizontal so both the water and the bucket shouldn't have a vertical velocity and they are falling at the same rate so how will they press against each other?

Consider this scenario. You are sitting in a car that for some reason is rolling backwards pretty fast toward a cliff. Superman lands on the edge of the cliff, and plants his feet. As the car approaches him, he reaches out and catches the car with his hands, smoothly slowing the car down, stopping it, and pushing it forward again in one smooth fluid motion. Your car is now rolling forward, away from the cliff. You are saved.

What does this feel like to you inside the car? As the car rolls backward you are sitting in your car, blissfully unaware of your peril. Then as Superman does his thing you feel gently pressed back into your seat, because the car is accelerating forward. The point I am trying to make is that even at the instant when the car's velocity is zero, when Superman has brought it to a complete stop and is now beginning to push it forward--even at that instant, you still feel pressed into your seat. Since the car is *accelerating*, you feel a force. It doesn't matter that the car's velocity at that instant happens to be zero.

In the same way that you can feel pressed into the seat even when the car's velocity is zero, the water can feel pressed into the bucket even when the instantaneous velocity of the bucket is sideways, because the *acceleration* of the bucket is downward

 

[madah12]

ok for the second part I feel it contradicts Newtons first law for there exist an instant where you are accelerating but have a constant velocity can you justify that?

 

[The_Duck]

I'm not quite sure what you're asking here. If you are accelerating then by definition your velocity is not constant in time. Your velocity might be zero, for only an instant: for example, if I toss a ball straight up into the air, its acceleration is always g [directed downward] until it hits my hand again, even at the instant when it is at its peak and its instantaneous velocity is zero.

 

[madah12]

uhm nvm I just didn't thinkwell, thanks for helping me

 

[The_Duck]

In my Superman example my intention is that the car only has a velocity of zero for an instant, because from the time he first touches your car Superman doesn't stop pushing until the car is rolling forward again. He is applying a constant force: by F=ma you feel a constant acceleration. Acceleration is change of velocity, so your velocity is smoothly changing during the whole time Superman is pushing. Your velocity doesn't become constant until you are moving forward comfortably and he let's go.

I might not understand what you mean by a "constant velocity." What do you mean by this that is distinct from "instantaneous velocity of zero"?

 

[madah12]

yes kind of mixed up the zero instantaneous velocity with the constant velocity so thanks for clarifying :)