Linear Restoring Forces

     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 displacements and 0 at the point of equilibrium, restoring the motion of the block. 

Displaced Spring, Displacement in Green

    Here, the spring is not to be confused with a sinusoidal function, as it is merely a representation of what a spring would look like, and doesnt reflect the shape with 100% accuracy. If we assume this is the maximum extent of the displacement on the spring (2 units, labeled by the green line), then at this point the restoring force is -k(2), accelerating the block in the other direction to cause oscillation and perpetuate harmonic motion. 

    To relate this to the previous blog post regarding waves (before having covered energy in SHM) we can assume the motion of the block is related to a sinusoidal function. In this case, displacement would be the output values for time intervals, for which the amplitude would be the displacement from the point of equilibrium to the maximum displacement. Also, we can describe the period to be the time for the block to go from one extreme and complete one oscillation back to the same extreme displacement. 

F = -kx

    There are some additional aspects of Hooke's law to note. First is that F = negative kx, as the restoring force causes motion in the opposite direction of the dispalcement to "restore". Next is that force varies directly with displacement with the spring constant (k) as the slope, defining a linear restoring force.

    In the case of hanging masses, the restoring force follows the same principles, except that equilibrium is achieved at a different point, as it requires opposition to the applied weight force by displacing the string.

Model of a Guitar String

    When guitar strings are plucked, and are captured in photo technology that captures the length of the frame from left to right over a constant period of time, it models a sinusoidal function. This is because points of displacement are mapped at their corresponding varying times, modeling a d vs t graph. The reason why this can be classified as simple harmonic motion is because a string functions like a spring in the sense that it sees a restoring force as a result of its displacement, perpetuating harmonic motion. The equilibrium point is at the center of the vibration (original position of string), amplitude is characterized by how wide it vibrates, and the period correlates with the frequency, and thus, the sound we hear. Pretty cool!

    With the example of a pendulum, forces on a mass for a pendulum are weight, and tension. For varying angles at which the pendulum swings, the linear restoring force is the component of weight that is not perpendicular to tension. Tension will always cancel out weight to maintain equilibrium on that one axis, so a restoring force contingent on an angle, similar to hookes law depending on displacement, restores motion. With small angles, -mgsin(x) can be approximated to -mgx, with constant values for m and g, and x varying to bring simple harmonic motion.