1 / 13

Modelling harmonic oscillators

Modelling harmonic oscillators. By the end of this presentation you will be able to: Model the motion of a mass oscillating between two springs. Predict the shapes of displacement/time, velocity/time and acceleration/time graphs for an oscillating mass on a spring.

jestine
Télécharger la présentation

Modelling harmonic oscillators

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Modelling harmonic oscillators

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

  3. Using SPRINGS. We decided that springs had a lot to do with Harmonic motion in the last presentation. Why? • They store energy. • You should have covered the energy stored in a spring already. • P.E. = ½ k x2 where k is the SPRING CONSTANT in Nm-1 and x is extension in m.

  4. 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. Constant We’ll use these relationships in our analysis of an oscillating mass held between 2 walls by a pair of springs.

  5. A graph of acceleration vs displacement: acceleration displacement

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

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

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

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

  10. 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.

  11. 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 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.

  12. 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

  13. 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?

More Related