Modelling harmonic oscillators

By the end of this presentation you will be able to: 1.Model the motion of a mass oscillating between two springs. 2.Predict the shapes of displacement/time, velocity/time and acceleration/time graphs for an oscillating mass on a spring. 3.Predict a force/time graph for the same oscillator. 4.Know the generic relationship for any harmonic oscillator. 5.Calculate the frequency of any mass oscillating on a spring from its mass and spring constant.

Using SPRINGS. We decided that springs had a lot to do with Harmonic motion in the last presentation. Why? 1.They store energy. You should have covered the energy stored in a spring already. P.E. = ½ k x 2 where k is the SPRING CONSTANT in Nm -1 and x is extension in m.

2. They “bounce” up and down in a unique way. Their “Flight plan” is well-known. a = - k/m x s where a is acceleration towards centre m is mass k is the spring constant in Nm -1 s is displacement from centre. “ - ” means: the mass is always being RESTORED towards the centre spot. We’ll use these relationships in our analysis of an oscillating mass held between 2 walls by a pair of springs. Constant

A graph of acceleration vs displacement: displacement acceleration

displacement acceleration The acceleration is always changing. This is new to us. Usually, it’s constant. (Commonly, it’s 9.81 ms -2.) What does this tell us about the RESTORING FORCE? (The force pulling towards the centre?) It’s changing, too

displacement acceleration And, most disturbing of all, where is the greatest acceleration? And where are the points where there the acceleration is zero ?

displacement acceleration Zero acceleration where velocity is biggest. Biggest acceleration where velocity is zero

+s- s - A +A +acc & +force - acc and - force V max V min a min a max

Let’s look a little more closely at our spring system, and use our knowledge from the previous presentation. Newton’s Laws lets us find acceleration, given the mass and the spring force at a given displacement s. We can find the change in velocity from the acceleration, From the new velocity, we can find the new displacement….. We can track the motion, moment by moment.

We know that s = A cos 2  ft if our oscillator starts from A. (see earlier presentation). Then v = ds/dt = - 2  fA sin 2  ft, using calculus Also a = dv/dt = - (2  f) 2 A cos 2  ft = - (2  f) 2 s a is proportional to - s It would seem that all harmonic oscillators obey the rule: a is proportional to - s Which indeed is the case. And there are lots of examples of them, as we shall see.

There’s a FINAL RESULT to this analysis. Remember that, for a spring, a = - k/m x s? Let’s use both expressions: a = -k/m x s = - (2  f) 2 s Cancel the – sign and “s”. (2  f) 2 = k/m or 2  f= (k/m) Note that T can easily be found from f=1/T

Try these questions: 1. What’s the spring constant of a spring with natural frequency 3 Hz and a mass of 2.5 g hanging from it? 2. What mass is required for a spring (spring constant 0.75 Nm -1 ) to vibrate with a frequency of 0.6 Hz? 3. What’s the natural frequency of a spring with spring constant 20 Nm -1 and a mass of 0.65 kg attached to it? 4.What’s the period for 1 oscillation for ex. 3?