Simple Harmonic Motion

 Energy, Connections between SHM and Uniform Circular Motion        What are some similarities we know of both SHM and UCM? They both repeat, they both have predictable motion, forces, velocities, and we know of the equations that describe both. For UCM, the acceleration is constant in magnitude, but changes direction to point toward the center of the circle always. For SHM, the force points toward the center as well, and scales with the distance from equilibium. SHM is modeled by sinusoidal functions and in 1 dimension, while UCM is in 2. Wait, thats it! Diagram Projecting the Circular Motion of a Rotating Ball           Projecting the "shadow" of the ball in uniform circular motion isolates the motion in 1 dimension, and whats left is the shadow moving in simple harmonic motion! When the force is pointed in the vertical direction, or has components vertically, it isnt contributing to the motion in that dimension, so it isn't refle...

      Because we have already covered (see previous blog posts) the causes of SHM and aspects of graphs related to SHM, we can now put the two together.      To set the foundation, we can use the mass hanging on a spring as an example. The position of the mass relative to the position of equilibrium can be modeled by a sinusoidal function, as proved in previous posts, and included below:  Position Time Graph     To recap, the amplitude is the average of the maximum and the minimum, the period is the time from one peak to the next, and the x-intercepts are points of equilibrium. Because the x values represent time, we can find velocity using d/t, or the slope of this graph. Because the slope is changing, we can measure average values for the slope over long intervals... Actually, we can't.     Using average values over long intervals means a resultant piecewise function. To get around this, we can decrease the time interval over whic...

      In any system at rest, forces drive motion (Newton's 1st). In the case of simple harmonic motion, the change in displacement isn't constant, and neither is the change in velocity. This means the force driving the motion has two qualities: it's nonzero, and its changing, maintaining simple harmonic motion. We call these forces "linear restoring forces", because they're responsible for restoring simple harmonic motion in a *linear* fashion.  Spring in Equilibrium     Hooke's Law states that F = -kx, which is a common example of a linear restoring force. Above is a model with a spring at equilibrium, meaning no force is applied on the block (at rest). However, when the block is displaced, an opposing force described by Hooke's Law exerts backward toward equilibrium. Additionally, because the spring force corresponds to displacement from equilibrium, as the block moves with the spring, the force varies in magnitude, being the greatest at the maximum ...

      As we discuss simple harmonic motions, or SHM, we will be dealing with many wave functions. Below is an example of a wave function. We first need to understand key points about this graph until we can fully understand SHM. Here is a graph that we can assume is a distance-time graph.     First, recognize that wave functions repeat themselves over equally spaced intervals. In the graph above, for example, we can see that it takes the same difference in the x-value to reach the first highest point of the graph to the second highest point. Conversely, it takes the same x-value for the first point of the lowest point to the second lowest point. The interval of which it takes the wave function to fully undergo one repeat is called a period , T. In SHM, it is measured in seconds.  When we talk about periods, they are not to be confused with frequency. Frequency is the inverse of periods, and describes how many times something happens in a second. So, the un...