Solved example: pressure needed to compress water

how much fresher is needed to compress water by 1% even bulk modulus of water is 2 Giga Pascal's alright let's solve this we'll start by collecting some data from the question um we've been given the bulk modulus we have been asked to compress water by 1% and for that how much pressure is required that's the question so based on this we could pretty much say that we're dealing with bulk stress and both strain right so the first thing we write down is the connection that we know between them and that's Hookes law so let's write down Hookes law Hookes law say is that bulk stress is proportional to both strain so both stress which is just the pressure is proportional to both strain and both strain is defined as the change in volume per unit volume so this is both strain and the proportionality constant itself is the bulk modulus bulk modulus and we've spoken a lot about this in the previous video so if you need a little bit of refresher on this or maybe you didn't revise this then it's better to go back watch that video first and then come back over here anyways when I will be dealing with problems on the elasticity with very tiny percentage changes here's a way which I like to think about this the first thing we'll do is get to understand this equation a little bit better get to understand the bulk modulus a little bit more intuitively so for that we could say that bulk modulus will be equal to pressure will be equal to pressure when this number becomes 1 when Delta V over V becomes 1 but what does it mean for Delta V or V to become 1 what does that tell us well Delta V is the change in volume and of course we're dealing with compression that means the changes in volume are going to be negative our volume will decrease so there must be a negative sign over here but let's not worry about that too much all right so Delta V is the change in the volume and V is the initial volume so when Delta V over V equals 1 we're saying the change in the volume should equal the volume meaning if the initial volume was 5 we come president by five if the initial volume was 10 we compress it by 10 so we are compressing it to nothing in other words we're talking about 100 percent compression so this means 100 percent compression and at first you might be like whoa can we do that can we compress something to nothing and the answer is no you can't do it but that's a nice way to think about it about solving problems so we could say the bulk modulus tells us how much pressure is needed to compress something by 100 percent which means for water we require 2 Giga Pascal's to compress it by hundred percent so now that we know that how much pressure is needed to compress by one percent oh we could just solve this doing a cross multiplication right that's why I like to think of it this way so let's go ahead and write that for water we now know that 2 Giga Pascal's 2 Giga Pascal's is needed for 100% compression gives us hundred percent compression and don't take this literally ok it doesn't literally mean that you can put 2 Giga Pascal's and you compress water to nothing remember Hookes law only works within elastic regions so this will only work for very tiny values actually but anyways this is nice very nice way to solve the problem right so do your best Dukey teruki gigapascals uses 100% compression so to get 1% compression how much pressure is needed oh we can just cross multiply and get the answer the pressure needed would be 2 Giga Pascal's times 1/100 and that pressure would be let's see Giga is 10 to the power 9 ok let's go ahead and write that we could say it's 2 times 10 to the power 9 and we cut 2 zeroes so you end up with 2 times 10 to the power 7 Pascal's and that's our answer that is the amount of pressure needed to compress water and now just to give you a little feeling for how big this pressure is we can write this a little differently I'll go ahead and write this as we write that over here everyone Marc you're here okay we'll write this as P equals 10 to the power 7 can be written as 10 to the power 2 times 10 to the 4 or 5 right we can do that so 10 to the power 7 Cambridge nice 10 to the power 2 times 10 to the power 5 and this is a hundred and 100 times 2 would be 200 and you might be wondering why I'm writing like this you'll see so this is 200 times 10 to the power of 5 Pascal's and guess what 10 to the power 5 Pascal's is the amount of pressure that we are experiencing right now due to the atmosphere so you may be sitting in your chair or maybe lying on your bed and watching this then there's air that's pushing on you and right now you're pretty much experiencing so the total pressure that you're experiencing due to air is roughly 10 to the power 5 Pascal's which means the amount of pressure needed to compress water just by 1% is 200 times that number that is an incredible amount of pressure needed just goes to show you how long compressible water is and now you'll agree with me that most of the times we like to think of water as a non compressible material and the reason is it's not literally non compressible you can compress it but even for a very tiny percentage change the required pressure is so humongous and we can pretty much assume water to be non compressible