in this video we're going to review simple harmonic motion we're going to start with a vertical spring when the vertical spring is not stretched it has a certain amount of length then when we put a mass on the spring it stretches downward and we're going to have a certain amount of displacement and we'll call this X1 if we put more mass let's say we double the mass we're going to get even more displacement and we'll go ahead and call this X2 if we graph it on on a force verse displacement graph we get a graph that looks like this we get kind of a linear graph here and the slope of this graph we have a name for it we call it K and this is the spring constant and the spring constant tells us how uh stiff the spring is if it has a large K uh then it means the spring is very stiff if it has a small k a small slope then it's telling us that uh the spring is stretchy now if you do damage if you overstretch the spring and you damage it it's not going to follow this graph so this is assuming a a undamaged spring that's following what we call hooks law using the force verse displacement graph we can derive a relationship for the restoring force and the displacement on a spring and we have a name for this uh relationship we call this hooks law and hooks law states that the restoring force is equal to the K which is the spring constant times the displacement and now the reason we have this negative is because the restoring force and the displacement are in opposite directions so on this spring right here we have a displacement that goes in this direction and so that is the displacement however the force that's that of the Spring Pulling on the box is in this direction going upward so they are in opposite direction so that negative is there because it's showing you that the spring force and the displacement are in opposite direction another thing that we can figure out from the force verse displacement graph uh of a spring is the area under the curve which represents the energy that's stored in the spring since this graph has an area under the curve that is the shape of a triangle we can use our triangle area of a triangle equation and so if you take a look at the unit for the Force which is Newtons and the unit for the displacement which is meters and if you multiply those two you get newton meters which is also a jewel so which suggests that the area on the curve um with a unit of jewels is representing energy and so here we have uh 1 over2 * the base time height uh the base uh is X the height is the force and we've already derived that the restoring force is equal to KX so we can substitute KX in here and we end up with 1 over 2 kx^ 2 our second method here is going to be using our work uh equation which is the force times displacement and so if you were to pull on the spring because the force is increasing linearly with the displacement um we're going to use the average force to calculate the work done because this is increasing linearly we can do that we can just take the average force and the average force is going to be 1 12 KX since the force is KX so it's going to be 1 12 KX * X and uh we end up with 1 2 kx^ 2 since work equals a change in energy if you do work on the spring if you pull on the spring it's going to store energy in the spring and the amount of work you pull on the spring is going to equal the amount of energy um increase in potential energy in the spring uh we can say that the elastic potential energy this is the energy stored in the spring is equal to 1 12 kx2 so whether you use the first method or the second method method uh you derive the same equation as you stretch the spring and pull it farther you have more energy stored in the spring next we're going to talk about two quantities that we can measure uh when we're looking at simple harmonic motion and so the first is a pendulum that swing back and forth back and forth and what we can measure is the time we can measure the time it takes for it to make one complete swing also a we have a mass here on a spring this is a mass on a spring on a horizontal spring and this mass will go move to the right to the left to the right to the left back and forth back and forth as it makes one complete cycle uh we can also time that and uh the time for one complete cycle is called the period uh so to calculate the period we can one way we could do it experimentally is to measure the total amount amount of time divided by the number of Cycles and that will give you the period and this is the time to complete one cycle also we can calculate the frequency the frequency is the number of cycles per second and so what you could do is you could um measure how many cycles the number of Cycles divided by the time to complete those cycle and that will give you the frequency now you'll notice that these look very similar except they're kind of inverse of each other uh so the period is equal to the one over frequency and frequency is equal to 1 over period so now we're going to analyze a harmonic oscillator which is a system that when displac from its equilibrium position experiences a restoring Force F proportional to the displacement X so this Mass here when I pull it towards the right um there's going to be a force on the mass towards the left and you can see the force diagram here on the right so there's an upward normal force downward force of gravity or weight and then we have the restoring Force towards the left once the mass reaches reaches the equilibrium uh there's no more restoring Force the spring is unstretched and now it just has a normal and the force of gravity acting on the Block when the Box continues moving towards the left now there's going to be a restoring Force toward WS the right because now the spring is compressed and it's going to be pushing the Box towards the right at the equilibrium the net force is once again zero uh the north uh the normal force is canceled out by the force of gravity weight and once it goes past the equilibrium we once again have a restoring Force towards the left when the block is at the amplitude um one thing you'll notice is that the uh for an instant the block is going to be at rest it's not going to be moving and so we say that the velocity is going to be zero the acceleration of the block at the amplitude is going to be maximum because that's when you have the maximum Force remember the more you stretch the spring the more restoring Force there is so at the amplitude you're going to have maximum force and maximum acceleration um because of newon Second Law fals Ma you also get the same thing once the uh block reaches the negative amplitude um zero velocity and maximum acceleration and then when it returns back to the uh positive amplitude you'll have zero velocity and maximum amplitude at the equilibrium where x equals z you're going to have maximum uh velocity and the acceleration will be zero and that makes sense because the net force uh of the block at the equilibrium is going to be zero so the net force is zero and f equals Ma so therefore acceleration will also be zero one of the interesting things about the period of a simple harmonic oscillator is that it's independent of the amplitude so it doesn't matter how far you stretch the spring or on a pendulum how far you pull it uh the period is independent of the amplitude so what what is it based on so for a spring it's based on the mass and the spring constant the period is equal to 2 pi * the square root of the mass divided by the spring constant since period and frequency are inversely Rel ated um if we take the inverse of the equation we can solve for the frequency using the mass and the pre uh spring constant so the frequency is equal to 1 over 2 pi * the < TK of k / M so you'll notice the equation is just the the inverse of the equation for the period now we're going to look for look at a simple pendulum where the Theta is less than 15° so if you have a pendulum where you are pulling it away from its equilibrium there will be a restoring ing Force um caused by the weight of the pendulum and it's going to be uh in a direction that is perpendicular uh to the string so we have a string that's pulling on this pendulum provides a tension force and um the component that's perpendicular to that the the component of the weight that's perpendicular to that um is going to be mg sin Theta and it's going to be negative right here because the restoring force and the displacement um are in opposite direction for Theta less than 15° sin Theta is approximately Theta and so uh the force is going to be approximately mg Theta and because Theta is equal to uh the displacement divided by or the Arc Length divided by the length of the string uh we can say that the force the storing force is going to equal to mg s / the length of the string and if we look at this equation this reminds us of hooks law FAL KX and uh the K here instead of the K here we have mg over l so the mg so K here is equal to we're going to set equal to mg over L um because this equation has the form of Hook's law and so by substituting mg over L for K we end up with that the period is equal to 2 pi Square < TK M ided mg / L and you'll notice that the M's will cancel out and this leaves us with the period equal to 2 pi Square < TK L will be on the top G will be on the bottom and so this is the equation for the period of a simple pendulum where you're pulling it less than 15° uh the period equal to 2 pi * the sare root of L over G so L is the length of the pendulum uh and G is just 9.8 so now we're going to cons look at the energy uh stored in this horizontal spring and assuming that there's no dissipated forces the total mechanical energy which is potential energy plus kinetic energy is going to remain constant so I have this block here and I'm going to pull it to the right and um when this this is where the spring is stretched uh and it's going to have stored energy so elastic potential energy so I'm going to go ahead and put a some elastic potential energy there and then as it gets pulled towards the left and it reaches the equilibrium that's when it's going to be the fastest so now I'm going to have the most kinetic energy there and then it's going to continue moving to the left uh once the spring is compressed at the negative amplitude we're going to have all potential energy now in between the amplitude and the equilibrium it's going to have some kinetic energy and some potential energy and then once it gets back to the equilibrium it's going to have all kinetic energy as it moves to the right the kinetic energy is going to transform into potential energy at the amplitude it's going to be all uh potential energy at the amplitudes the potential energy stored in the spring is going to be 1 over 2 Ka a^ 2 it doesn't really matter if it's negative amplitude or the positive amplitude because it's just going to be the magnitude and then at the equilibrium the kinetic energy will be 1 over 2 MV uh Max squar that's where it's going to be moving the fastest and so we're just using our kinetic energy equation and often times you'll be setting these equal to each other uh to find the unknown for example you might be looking for the uh maximum velocity and you're given the amplitude the mass and the spring constant