Ideal spring and mass system Simple harmonic oscillations

Ideal spring and mass system. Simple harmonic oscillations. Force. Energy. Amplitude. Phase. Combination of several springs (in series and in parallel). Professor Marko B Popovic

Ideal spring and mass system Spring constant Stable equilibrium point, X=0

Ideal spring and mass system Spring constant Stable equilibrium point, X=0 Restoring force

Second Newton’s law For 1 dimensional case, motion along x only Hence

Second order differential equation or Solutions for equation are

Let’s check that Solutions for equation are

Because acceleration in 2 nd Newton’s law is second derivative of position most of classical physics problems can be expressed as 2 nd order differential equation. To completely solve this equation one needs two boundary conditions, i. e. essentially two extra specifications of problem in hand to determine two undetermined constants C. For example, for – sign differential equation we just addressed Actual solution 2 nd Newton’s law Math General solution Boundary condition

Back to our spring mass system, equation is and solutions are Example of Simple Harmonic Oscillator angular frequency

actual solution

actual solution

amplitude phase

Other physical systems behave similar to spring mass system. Consider “mathematical” pendulum. and 2 nd Newton law for rotations

But we already know solution for this type of equation

Consider two springs Connected in series m

m or

Consider two springs Connected in parallel m

Let’s talk about energy

Energy