# Pascal's Principle It's common knowledge that liquids and gases will take the shape of the (closed) container they're located in - though a gas has no fixed volume while a liquid does, meaning that a gas would spread out to fill the container. This fixed volume means that gases are compressible while liquids aren't (in general). If you've been wondering how a hydraulic press (of all things) works, it's with this principle! For context, you might've seen one of these before - where someone pushes down on a tube and an object typically too heavy to lift is pushed up, perhaps a weighted stuffed toy making its way onto a conveyor. ![[Pasted image 20230828195430.png]] This is a quick diagram of a hydraulic press. When we apply a pressure to it, $p_{inp}$, the water level on the open face increases. As $\rho gh$ in calculating a liquid's density does not change, the value for the change in pressure is: $\Delta p = p_{i}$ Ever watched the video Delta P? It's great - and it's got something like this too. Since this change in pressure happens within a closed system (there is only the liquid) the open face has to increase by a proportional amount - the change in the pressure on both sides is equal. Hence the small surface area - it's much easier to compress an area with a smaller surface area - see the pressure formula! $\Delta p = \frac{F_{i}}{A_{i}} = \frac{F_{o}}{A_{o}}$ Let's get the output force (the upthrust caused by compressing the water) giving us: $F_{opt} = F_{i} \frac{A_{o}}{A_{i}}$ This means that the principle basically shows we can use liquids as direct force multipliers - whatever the input force was has now been multiplied by the ratio of the areas.If the water level on the left decreases by a certain amount $d_{inp}$, this value will equal the surface level increase on the right (the larger volume). As there isn't any new liquid being added to the system, we can get an equation: $V = A_{i}d_{i} = A_{o}d_{o}$ To find the output distance, we just have to rearrange the equation: $d_{opt} = d_{inp} \frac{A_{inp}}{A_{opt}}$ These two components - $F_{o}$ and $d_{o}$, can be multiplied together to find the work done of the object: $W = \left( F_{i} \frac{A_{o}}{A_{i}} \right)\left( d_{i} \frac{A_{i}}{A_{o}} \right) = F_{i}d_{i}$ So the next time your parents crush your electronics under a hydraulic press, make sure you let them know what physics has to do with it! You might even be able to convince them to buy you a new laptop... >[!Example]- Problem Taster > # Archimedes' Principle *Prerequisite for* [[Buoyancy (Mechanics)|buoyancy]] *among many other things. Please! Pay! Attention!* Simply put, this is a "domesticated buoyancy" tab - all it is is a glorified recap of [[Newton's Laws and SUVAT#Newton's Third Law|Newton's Third Law]]. We'll be going over the basics of buoyancy - though there's an unfortunate lack of mathematical intuition which you can make up for [[Buoyancy (Mechanics)|here]], if you dare... (unfinished intuition in part thanks to no multivariable calc knowledge) So, imagine there's a big container of water and we're going to try to sink a pillow in it. Leaving behind water absorption and other things, we can try to imagine it as just sinking downwards thanks to its weight $mg$. Eventually, this pillow will suspend itself within the water. Why is that? To understand, let's take another look at terminal velocity - where the action force of the object (in this case, the weight) is equal to the reaction force of the object (the upthrust). In this special case, we can call the upthrust the **buoyant force** - being equal to the weight of the object as shown: $F_{b} = mg$ This is what makes the objects in your pool seem lighter than they actually are! Note that since this force acts opposite the weight of the object, we can create an expression relating $W_{app}$, the apparent weight, and $W_{abs}$, the actual weight: $W_{app} = W_{abs} - F_{b}$ Let's generalise this into [[Newton's Laws and SUVAT#Newton's Second Law|Newton's Second Law]] by using the identity $m = pV$: $p_{w}Vg - p_{obj}Vg = p_{obj}Va$ >[!Success]- Not-So-Buoyant Objects >Objects can suspend themselves underwater - >[!Example]- Problem Taster ## Floating - A Fluids' Conundrum Let's define what a floating object really is! When an object floats, part of it is going to be submerged under whatever medium you're using. ![[Pasted image 20231003082043.png]] **In this case, the weight of the volume of water we've displaced is going to equal the buoyant force:** $W_{d} = F_{b}$ We're done! This wasn't too difficult - though do remember the definitions posed in this. Pretty(very) important in fluid dynamics! >[!Danger]- >[!Example]- Problem Taster # Example Problems # What's Next? One last hurdle in fluids - Bernoulli's Equation, which describes how a fluid flows in an expanding pipe: [[Bernoulli's Equation (Mechanics)]]