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We just started learning about pressure in our school, and i am a bit confused. I tried to google extensively, but i didn't really find much.

In mechanics, so far in our school we've been taught about Newton's laws of motion and some related stuff, and suddenly we started learning about pressure. I asked this same question to my physics teacher, but he failed to answer.

How does pressure actually factor into mechanics? We studied so far that to cause motion, you require a force. Then we study about pressure, and the examples are ones like pressure of gas in a contained beaker and then lowering/raising it's volume. Suddenly in the applications section however there are things like Paper pins have a low surface area at the end to maximize pressure, and that you can't walk on sand easily because the sand depresses under pressure. I understand the math behind it, but what does the math actually mean?

I mean, a force is being applied. It should cause motion, but apparantly if the surface area of the end is small then pressure goes up? The force here remained constant(?) so how does pressure factor into the mechanics? Why did that just happen?

I realize i may be way off topic and asking a stupid question, but i am having some trouble comprehending how pressure works. I understand the examples of things like pressure in a container, atm. pressure e.t.c but how does pressure transmit in a case like this? My apoligies if this is a stupid question.

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A force is a vector. It represents a value and a direction. Pressure applied on a small surface results in a force applied in the direction perpendicular to the small surface and with value $F=P\, A$ where $P$ is pressure and $A$ is a small area.

If you have an object, where low pressure is applied on a large area on one side, but high pressure is applied on a small area on the other side the result can be two equal and opposite forces which will keep the object static (motionless). $ P_{small} A_{large} = P_{large} A_{small} $

On the contrary in a hydraulic lift it is the pressure that is preserved in the fluid, so if a force $F_{in}$ is applied on one end with area $A_{in}$ then on the other side the force $F_{out}$ over the area $A_{out}$ is found by the relationship

$$ P = \frac{F_{in}}{A_{in}} = \frac{F_{out}}{A_{out}}$$

So to apply a large force on the end you need a large ratio $\frac{A_{out}}{A_{in}}>1$ to get

$$ F_{out} = \left(\frac{A_{out}}{A_{in}}\right) F_{in} $$

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There is no stupid question when it comes to genuinely trying to understand physics. Pressure is the result of molecules smashing against the walls of their enclosure. Except at zero Kelvin, there is always some kind of erratic movement which increases with temperature. For a given number of molecules in an enclosure, the smaller it gets and the more impacts the molecules have on the walls from their movements. Imagine you're in a crowdy room and everyone is pacing back and forth, more when it gets hot, and the walls get smaller like in star wars. Or more people are coming in. For air in a balloon for example, the walls are a flexible membrane and can extend if the pressure (how much the molecules hit the walls) increases. In most cases where you deal with noble gases or low pressures, you can derive everything from the perfect gases law. For a given pressure, the more surface it acts on and the more force you get (P=F/S), just like having more people pushing a car - but usually as a result it takes more energy to pump that pressure up as the volume to fill is bigger. The piston is the best example for that.

Edit: StickyCube made me realize I did not specify that pressure did not have to be of a gas. The knife is an excellent example, because in mechanics the most common materials failure criteria is in units of pressure: force does not really matter, pressure does. I once read in an issue (can't remember which number, the one about the most impressive things ever manufactured) of "How it works" that the biggest bucket-wheel excavator ever made, weighing 13000tons, not only was able to move but would not hurt at all a worker that would have his feet underneath the treads as the tread surface is so massive.

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In physics, pressure, $p$, is a Force, $ F $, per unit Area, $ A $. Where:

$ p=F/A $

We can see from the simple formula that to increase pressure we can increase the force or decrease the area.

The example which you gave of a paper pin demonstrates mechanical pressure, by lowering the surface area of the end of a pin we can more easily penetrate the paper because the force that we exert is directed on a smaller point. Similarly, this is the same reason that knives have a serrated edge, to apply more pressure on the area that you want to cut.

Maybe the most common use in physics for the word 'pressure' is when talking about a gas. But what does the pressure of a gas actually mean? Every time a particle of a gas collides with the wall of its container, it exerts a force (due to the change in momentum). So when we have many particles all colliding with the container we can take an average of the force it is exerting and find the pressure. The mathematics of this is quite involved and is dealt with by a branch of physics called 'Statistical Mechanics' which derives bulk properties (like temperature or pressure) from systems which are made up of many bodies (like a gas).

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