 I love this drinking bird. With just a little bit of water in the cup, it'll move back and forth in a repeatable motion all day. This bird is an example of something that undergoes simple harmonic motion. It's a repeatable back-and-forth motion driven by a force. There's lots of different examples, but the one we're going to take a look at in this video is a mass on a spring. Imagine a frictionless surface with a mass like this box resting on it. Attached to the mass is a spring, which is also fastened to the wall. The spring has a particular elastic constant called k, which is a measure of how much force is needed to stretch or compress the spring by one meter. The mass and the spring start off at equilibrium. At this point, the spring is neither stretched out or compressed, it's just at its natural position. Equilibrium is shown as that little vertical dotted red line. What would happen if we applied a force and pulled the mass spring system to the right? The spring would stretch out by a certain amount. This stretch is called the displacement of the spring, and it gets the variable x. The maximum displacement from equilibrium has a special name as well. It's called the amplitude. The amplitude is the black vertical dotted line. There's one on either side of the box. Right now, the mass is at rest. As we release the mass, it starts to accelerate back to the left. Why does it accelerate? Because a force from the spring is being applied to it. According to Newton's second law, any net force causes acceleration. Now, when it comes to a mass in a spring, we're really concerned about three positions. Far right, center, far left. Let's take a look at what's happening there. When the mass is on the far left hand side, its displacement, x, is at a maximum. The largest it can possibly be. It's at its amplitude, but the amplitude is negative because it's moved to the left of equilibrium. The force is at a maximum as well, and that's because of Hooke's law. Hooke's law says the force in a spring is equal to negative kx. So if x is as large as possible value, then the force pushing from the spring back to the right has to also be at as large as possible value. The acceleration is at its maximum because of Newton's second law. If force is at a maximum, then acceleration has to be at a maximum as well, and the velocity is at zero because it's stopped. Let's go to the center. x is zero. The displacement is zero because we're right at equilibrium. The force from the spring is zero as well because Hooke's law tells us force equals negative kx, and if the value of x is zero and you substituted it into the formula, you'd end up getting a force of zero as well. The acceleration here is zero, and that's because in Newton's second law, you get acceleration from mass and force. And so if your force is zero and you put that into the equation, you get an acceleration of zero too. The velocity is at a maximum though. That's because all of the energy from the spring has been dumped into the box, giving it the largest possible velocity. Lastly, let's look at the right-hand side. The value of x, the displacement, is at a maximum. It's positive because it's to the right of equilibrium. The force is at a maximum because the spring is stretched out the most it can be right now, but it's negative because it's going to pull the mass back to the left. The acceleration always follows the force, so it's also negative and is going back to the left. And the velocity at this point is zero because the mass is stopped. When we see it all at once, we start to notice some patterns. The velocity is always the odd one out, and the force, displacement, and acceleration always seem to match up with one another, either maximum or zero. Although the directions of the forces and the accelerations and the displacements might change depending on where you're at. Knowing where in the movement of this mass each of these values is a zero value or a maximum is really important when we go to start calculating all of these values. This is a good time to mention that all the ideas we're looking at here as well as all the equations we're going to get into also apply in the same way to a vertical mass spring system. Here's how we find the maximum acceleration of the mass. It's a great review of net force statements. So our net force in this situation is made up only by one force, and that's the force in the spring. So net force equals FS force in the spring. In place of net force, I put an MA, and in place of force of the spring, I put in Hooke's law, negative kx. If I rearrange this, I get a formula that can allow me to find the maximum acceleration of any mass on a spring, vertical or horizontal. The acceleration is equal to negative kx over m. Now we can find the maximum speed of the mass. I know that when the mass was at one of its two amplitudes, all of the energy was the energy stored in the spring. It was elastic potential energy, and that that elastic potential energy changes into kinetic energy when the mass moves through its equilibrium point. So I can say EP, elastic potential energy, equals EK. In place of elastic potential energy, I can put in one half kx squared, and in place of kinetic energy, one half mv squared. The one halves cancel out, and what I end up with is a nice little formula that I can manipulate in a lot of different ways. I can solve for the speed, or I can go through and solve, if I know the speed, for spring constant, displacement, or mass. This is a very versatile little relationship that you can use to solve a lot of different types of problems. When it comes to finding period, there's no real theory to it. There's just basically a set equation you're going to use. The period of a mass spring system is equal to two pi times the square root of m over k. Period is the amount of time it takes for the mass to go all the way to one side and come all the way back. The time for one full back and forth movement. Using these three equations, we can analyze the physics of any horizontal or vertical mass spring system, which is a great example of simple harmonic motion. Check out some of these other videos about simple harmonic motion, and for more help on the topic, check out my website,