What is physically going on when I stick my finger in a glass of water and the scale tips downards implying it got heavier?

Question

Suppose I have a scale with a glass of water on one side and a counterweight on the other side. If I stick my finger in the glass of water I observe that the water side of the scale becomes heavier and the glass lowers.

My question is: what specifically is happening here? I'm not asking about the abstractions of buoyancy and normal forces, but what is physically going on at the molecular level that causes the glass side to dip down?

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From my understanding: The water in the glass is a "sea" of water molecules attracting and repelling each other. But they're not repelling each other too far because they attract more than they repel. As I'm pushing my finger into the glass, the molecules get pushed in the direction they get hit by the finger and creates a domino-like chain reaction of molecule collisions. The chain of collisions eventually reach the boundary where the glass meets the water which results in water molecules hitting the glass more frequently and at a faster speed. I can see that this increased amount and faster collision rate would push the glass down since it's transferring all that motion energy. Once the finger is at rest in the glass, I notice the glass is still pushed down by the finger.

This is where I'm not entirely sure what the correct explanation is.

Is it the case that: Since the finger is in the glass, the water level rises. Since the water level has risen, more water is touching the glass which means there will be more collisions of water molecules with the surface of the glass which will result in more motion energy transferred from the water to the glass which results in additional movement of the glass downwards.

Or: Since the water is displaced higher up onto the glass, there is more force from gravity pulling the water molecules downwards which then sends a chain reaction of collisions downwards towards the bottom of the glass. The overall effect is that the molecules at the bottom of the glass now are hit more frequently and at faster speeds due to the downward chain of collisions caused by gravity and this results in more and faster hits on the glass.

Or is there a more specific explanation about what happens? Again, I am specifically not asking about the law based explanations (newtons third law, etc.), just the process based explanation.

Answer 1

By placing your finger in the water, you are exerting a force and the water responds to that force by rising higher in the glass. That force adds to the apparent weight of the water/glass system. If you were to mark the place where the water rises on the glass and then remove your finger the weight would return to the previous value. Now if you pour more water into the glass so that the level returns to where it was with your finger inserted you will find that the weight has increased to the same value as before (with finger inserted). As you can see this has nothing to do with buoyancy but just with the volume of water displaced.

Answer 2

I'm not sure whether the previous answers address your interest in the molecular understanding of the change . The stress tensor tells us that pressure is momentum flux. The kinetic flux hasn't changed, but the intermolecular force term has. The force varies very quickly with distance and the molecules are a tiny bit closer together so they push harder on each other and on the glass. So the answer is that the water molecules are closer together and exert more force on the glass.

We normally think of water as effectively incompressible, but this prevents us from understanding the physics of certain situations, of which this is one. The pressure has increased and therefore the density has increased a tiny bit, giving the above explanation of the change on the molecular level

Answer 3

At a macroscopic level there is a buoyancy upthrust acting on your finger, the reaction to which causes an increased force on the scale. At a microscopic level the effect is no different to putting a floating object in the glass or even adding more water. The gravitational forces on every individual molecule of substance in the glass- whether that is just water, or water plus some floating body- act downwards. The weight of molecules higher in the glass puts pressure on molecules lower in the glass and in turn they put pressure on molecules of water below them, until eventually the molecules of water at the bottom of the glass apply pressure to the molecules in the glass itself, and so on until the lowest layers of molecules in the glass impose the cumulative force to the uppermost layer of the scale.

Answer 4

The scale tips because you are pushing down on it with your finger. It isn't as obvious that you are pushing on it as it would be if you touched the scale itself. You are pushing on the water, and the water pushes harder on the scale as a result.

The first thing you need to understand is that it doesn't matter how massive your finger is. The size and shape matter. We can simplify a bit if you use a cylinder instead. We will choose a weightless cylinder so that the force of gravity on the cylinder does not confuse the issue. We will also suppose the glass has no mass.

Such a cylinder would float on the surface of the water, not penetrating the surface at all. It would not change the scale. At this time, the water is motionless. The total force on the water is $0$. The two forces on it are gravity and the upward reaction force from the scale holding it in place. These are equal and opposite.

You can make the cylinder penetrate the surface by pushing it down. The water adjusts by rising up the glass a bit, and then stays still. There is an equal and opposite force from the water pushing up on the cylinder.

Now there are three forces on the water, but the water is still motionless. The total force is still $0$. The two downward forces are the force from the cylinder and gravity. The upward force is the reaction from the scale. The reaction force must be bigger now to balance both forces.

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From this you can see that it must work out that the downward force from the cylinder must get transmitted to the scale. But how does that happen? To see that, we will look at just the glass of water, and divide it into horizontal layers.

Let us start with one thin layer of water in the glass. The weight on the scale is the weight of the layer. The bottom of the layer presses downward with that much force.

Let us add a second layer. One way to think about it is that twice the water has twice the weight, and so it presses on the scale with twice the force.

Another way is that the top layer presses on the bottom layer with the weight of one layer. The bottom layer presses on the scale with its own weight, but also transmits the weight of the top layer to the scale. So a liquid can transmit a force, even though it is fluid.

You can continue this way with $3$, $4$, or however many layers you like. In each case, the force on the top of a layer is the weight of all the layers above it. The layer adds its own weight to this and presses on the next layer down. The force is proportional to how far below the surface the layer is.

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Let's do it again, a little differently. Pour in the first layer and hold the cylinder so the bottom rests on the surface. Nothing new here. The force on the scale is the weight of one layer.

Now pour in enough water so the water rises to a depth of two layers. So what is the difference between this and two full layers? There is a missing plug of water where there is now cylinder. And the force from the cylinder is just enough to keep the water motionless. it is just enough to keep the water from pushing the cylinder upward out of the water. That is just what the weight of the missing plug of water does.

This is key. The downward force from the cylinder must be the same as the downward weight of the missing plug of water. Both are just enough to keep the layer beneath the cylinder motionless. If you push harder on the cylinder, it will sink deeper where the upward force is bigger and will balance it. If you press less, the water will force it upward to where the upward force is smaller and will balance.

Water pushes on the sides of the cylinder as well. The cylinder must push outward enough to resist, or it will be crushed. Again, these forces are the same as those from the missing plug of water, which also must hold back the water around it. These forces on the sides are horizontal, and do not change the downward force on the scale.

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So that is what is going on macroscopically, without considering that water is made of molecules. But it doesn't really change anything when you think of molecules.

If molecules were motionless, like little grains of sand, the same explanation would work.

Molecules are really bouncing all around. All the bounces at the top of a layer kick hard enough to hold back all the layers above it. The layer below must kick a little harder to hold back those layers and one more.

There are differences between a gas, liquid, and solid. In a gas, the molecules will travel until they hit something.

In a liquid, they stick together but flow. They vibrate and don't travel away from their neighbors. But they and their neighbors will flow close enough to be in contact with the cylinder.

In a solid, they just vibrate in place. They kick whatever they are in contact with.

A higher temperature means more violent kicks. In a gas, this means molecules kick harder. The forces go up. In a sealed container, raising the pressure by raising the temperature does not change the weight. The gas kicks just as much harder on the ceiling as the floor.

In a liquid or solid, the forces do not go up with temperature. Atoms and molecules stick to each other. If a higher temperature means they vibrate harder, their neighbors hold them back harder.

Answer 5

My take on this is that what you are experiencing is buoyancy. The your finger’s displacement of the water creates an upward buoyancy force on your finger. Since you are a stationary and relatively immovable force, there is instead a downward force on the water. I suspect (without any factual basis) that if you performed a buoyancy calculation, the increase in the container’s mass would equal the same result as your buoyancy calculation.

If you placed a floating object in the water and added weight to it until it displaced the same amount of water your finger did, I suspect you also end up at the same resultant mass on the scale.

Answer 6

I'm missing something here. There's some sort of subtlety I haven't gotten yet.

Imagine the glass is not full of water but is full of air. A weightless piston can compress the air. It's completely obvious that the more you compress the air, the more the scale will say it weighs. You are literally pushing down on the scale, and the air pressure in all directions is pushing in one direction among the others, down.

Imagine the glass is not full of water but is full of, well, glass. If you push on the top surface of the glass it will push on the scale and the scale will say it weighs more. No question.

Why is water different? Water is a liquid. If the weightless piston pushes on it, then of course it will transmit the force like a gas. But if the piston has a hole in it, then the liquid isn't like a gas. It flows through the hole and not much happens. The gas would escape, and provide pressure in all directions -- to the top and the bottom of the glass, the top and bottom of the scale, etc. The glass will just sit there and its weight will affect the scale.

But now connect a 1/4" glass tube to the hole in the piston, one that's 3 feet long. Push on the piston and raise the water level up to 3 feet in a very small volume, with a very small area at the top. Now the scale WILL record something heavier. Nothing to do with the volume of water, just with the depth, the pressure. Isn't that how a hydraulic lift works?

Now pressurize either the gas or the liquid with the piston. Glue the piston to the glass wall, so you don't have to keep pushing it down. The pressure is the same. But the scale will give the lower weight. Won't it?

It isn't just the pressure. It's something else.