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so say you had a mask connected to a spring and that Springs tied to the ceiling and you give this massive kick well it's going to start oscillating oscillate down and up and down and up but if I were to try to draw this all those drawings would overlap it look like garbage forget that let's just get rid of all this let's say we just drew what the masses position look like every quarter of a second so this is what started that wait a quarter of a second now it's here quarter of a second now it's here so this is one mass and we just took pictures of it and we move those pictures right next to each other if I were to connect the dots I get a graph which is basically the height of this mass as a function of time it looks like this so if I were to plot this legitimately on an Y or vertical position versus time graph you'd get something like this it starts in the middle and it goes up and then it comes back down and this process repeats and you know what this is this looks like a sine graph but here's my question let's say we didn't start the mass in the middle and give it a kick upwards let's say at T equals zero we start the mass all the way at the top when we let it drop in other words we do this we start the mass up here we let it drop the graph would look like this this time if we were to plot this over here we'd get a graph that looked a little something like this it would start at the top then it would drop down it reach some lowest point it would come back up get up here this process would repeat over and over and over again my question is are these graphs the same or are they different well they're obviously different but almost everything is the same their amplitude is the same you know they have the same displacement from the equilibrium that's the amplitude that's the same their period is the same the period is the time between oscillations so the periods are both the same the only thing that's different is that one graph is shifted compared to the other in fact check this out if I were to take this green graph the initial one and just shift it left we get the exact same graph they're almost exactly the same one is just shifted and so the word physicists used for this idea the two graphs can differ by the amount one is shifted is the idea of the phase we would say that these two oscillators are out of phase how out of phase will one whole cycle would be from here all the way to there they're not that out of phase they're just shifted by this much and that it turns out for this case I've drawn here is only a quarter of a cycle so you can say these are a quarter of a cycle out of phase or if you think of the unit circle we know one quarter of a cycle would correspond to ninety degrees either ninety degrees or PI over two that's what these are these are out of phase by PI over two radians or ninety degrees so how do we describe this idea of phase mathematically well if we were to try to write down an equation for this green oscillator say at T equals zero it started there I might write down that okay Y as a function of time the height of this oscillator as a function of time would be the amplitude times since this is starting at zero I'm going to use sine because I know sine starts at zero when T equals zero of 2pi over the period times the actual time this little T is a variable this little T represents the actual time at a given moment so let's make sure that this equation actually works if I were to start off at T equals zero and plug in T equals zero and here the sine of zero is just zero so this whole thing becomes zero and that's what I should have I should start at T equals zero at a y-value of 0 so that's good and then as T gets a little bit bigger this inside amount gets a little bit bigger the sine of a tiny positive amount be a tiny positive number so that's why this graph goes up from there eventually it gets to the peak where will it reach its peak it will reach its peak when it gets through a quarter of a cycle remember a whole cycle is this whole amount here so a quarter of a cycle is just when it gets to the amplitude from zero and that will be at a time T equals the period over four and the reason is that's a quarter of a cycle does that met this is math actually give us that it does because watch if I plug in little T the time variable as the period over four I'll get the sine of 2pi over the period times the period over for because that was my time the periods cancel out I'd get sine of two pi over four is PI over two and PI over two that's 90 degrees sine of 90 degrees or sine of PI over two radians that's just one that's the biggest that sine can be that one times the amplitude is going to give me a value for the height of the amplitude this is going to describe my oscillator perfectly since it's going to give what the height is at any given moment of time so that wasn't too bad but what do we have to change in order to describe the purple oscillator so now I want to describe this purple oscillator you might say oh well easy just make this sine a cosine and yeah for this case turns out that works but pretend like you didn't know you could do that or if you want to make it harder say this was not shifted a perfect quarter of a cycle maybe it was only shifted like a ninth of a cycle then cosine is not going to do it for you you need a more general way to adjust how much this wave is shifted and that's going to be some sort of phase constant in here where do we put the constant you might think all right if I take my green graph I want to shift it to the left you might think well should I subtract some like amount out here that's not going to work that's going to take your whole graph and subtract from the value you get for the height a constant amount every single time because of that that will just take your graph and shift it downward that's not going to work if we added a value of V up here that's also not going to work that would just end up shifting our graph upward all right this B bet this B value is not going to do it for us so you might want that in certain situations that's not that's not going to shift the graph left or right turns out what we have to do shift it left to right is add a constant within this argument of the sine here so if I wanted to describe my purple oscillator I'd say that Y is a function of time is the same amplitude let's just again use sine it would be 2pi over the period times time and then I here's here's where the phase constant comes in I'd have to actually add a phase constant you might think subtract you might think subtract because we wanted go left turns out adding a phase constant will shift the graph left this was counterintuitive to me it's always freaked me out I'd always forget this how come adding a phase constant shifts the graph to the left well watch if we take this equation now instead of putting Phi in there this is the this is the symbol we use for the phase constant in general Phi and in this case we know what it should be it should be a quarter of a cycle and for a sine graph a quarter of a cycle is PI over two so let's just see if this works at T equals zero we used to get zero which is what we wanted but now for the purple graph I need to start at a maximum value so at t equals zero this whole amount right here becomes zero and I'm just left with sine of PI over 2 and sine of PI over 2 is 1 that's a maximum value so times amplitude would give us the amplitude which is what we want we want to graph that starts at the amplitude and this is better than just putting cosine released is better to know about because now even if this phase shift was PI over 4 or PI over 9 or PI over 27 you could shift by any amount you want using sine or cosine now you know how to shift these things adding a phase constant will shift it to the left subtracting will shift it to the right and the larger the phase constant the more it shifted you don't ever really need to shift it by more than 2 pi since after you shift by 2 pi you just get the same shape back again so this constant in here it's PI over 2 in this case in general it would look like this you'd have some oscillator it's got some amplitude you could use sine or cosine plus a phase constant and this phase constant will determine how much this oscillator is shifted left or right and I should say be careful physicists can be sloppy here and used the same word for multiple things sometimes the word phase is used just for this little part here this little added constant part but sometimes by phase people really mean this whole thing inside of here this whole term that you're taking sign of because this is what's determining where you're at in your actual cycle and these ideas don't just apply to a mass on a spring you could write down the equation for a wave you'd have an extra term in here for space not just time and guess what there'd be a little constant at the end that you could add that would be the phase constant so this idea of phase gives you a way to describe how two oscillators or two waves are shifted with respect to one another and it lets you account for all kinds of properties like we had for this mass on a spring