4.5 Elastic potential energy and Simple Harmonic Motion (SHM)

How does a rubber band reflect the link between energy and forces? Think carefully about the behaviour of stretchy objects like rubber bands and springs What happens to them?

Stretchy stuff responds to forces by storing energy When you apply force to an elastic object like a spring or elastic band The force does work on the object because it causes a displacement This gives the spring energy The spring releases this energy when it is given the chance to return to equilibrium length http://www.sciencewithmrnoon.com/physics/weightlab.swf

Fe Fg Fg Fg Fe Fg Fe Fe V = 0 m/s Vmax V = 0 m/s Equilibrium line without mass Equilibrium line with mass Fe Fg Fg V = 0 m/s Vmax Fg Fe Fg Fe Fe V = 0 m/s

How do we know it releases energy? Well, what were to happen if you placed something in front of that spring?

Elastic potential energy Energy stored in springs are a type of potential energy Like gravity, once the energy is put into the spring, it can be released if the conditions are right In fact, elastic potential energy is similar to gravity You can compare pulling on a spring to Eg

In gravitational potential energy, stored energy from raised objects is released when the gravity pulls the object back down – similar to what the spring does when it “pulls” the mass back to equilibrium point.

Ideal spring An ideal spring is one that doesn’t deform when stretched or depressed That means it doesn’t get damaged and can return back to normal shape An ideal spring that we study also assumes that external forces like friction do not interfere with it – nor does the spring experience any internal forces

Reality vs. ideal conditions What happens eventually to a bouncing mass on a spring in real conditions? If you had an ideal spring, what happens eventually to the bouncing mass?

Hooke’s Law An ideal spring follows Hooke’s Law, which states that the force required to deform a spring per unit length is always constant Where: F = -kx F = force applied in Newtons (N) k = spring constant in N/m x = position of spring relative to equilibrium in metres (m)

Why the negative? The negative is supposed to make up for the relativity in direction of forces Hooke’s law is written from the spring’s point of view The F value in the equation refers to the force that the spring is applying on the mass

Hooke’s Law is written from the point of view of the spring

Simple Harmonic Motion SHM is created when the force (and therefore acceleration) is proportional to the displacement In the previous animation of the spring’s motion, notice that the net force on the mass is the greatest when the displacement is greatest from equilibrium

Fe Fg Fg V = 0 m/s Vmax Fg Fe Fg Fe Fe V = 0 m/s

SHM creates periodic motion This relationship is periodic – like waves, it exhibits regular, repeated motion that can be described using many of the same properties that you use to describe wave functions Imagine the spring that we discussed earlier, but this time it moves along a track The mass is attached to a writing device that can sketch out the path of the mass

Mathematically… In order to make a clear relationship between the spring and the forces associated with it, you can compare the movement of the bouncing mass to a handle on a rotating disc

Direction of rotation

Direction of rotation

Direction of rotation

Direction of rotation

Direction of rotation

Conservation of energy still applies If you analyze the motion of a mass on a spring, the speed gained by the mass and lost by the mass is traceable back to the total amount of elastic potential energy put into the system A spring with a mass that is stretched or compressed will store energy Once released, the mass will move As the kinetic energy of the mass increases, the energy in the spring decreases, and vice versa

Dampened Harmonic Motion DAMPENED HARMONIC MOTION occurs when the spring system loses energy over time causing the displacement to decrease in the spring This energy is dissipated to other forms This is desirable in some mechanical systems like shock absorbers – or else your car would continue to bounce up and down after the shock absorber is depressed