When we talk about Simple Harmonic Oscillators, we will describe their motion with two primary characteristics, frequency and Amplitude. Which of these can be changed merely by changing the “initial conditions,” usually by just pulling or tugging on the system harder to set it in motion and which can be changed by altering the physical components of the system, mass, spring constant or length or string, etc. • Amplitude is changed by either method • Frequency is changed by either method • Amplitude is changed ONLY by how hard you tug, frequency is changed ONLY by altering the system • Frequency is changed ONLY by hard you tug, amplitude is changed ONLY by altering the system.
Correct Answer – C If you take a mass on a spring or a simple pendulum, you will find that the frequency of the oscillations is completely independent of how far you pull back the mass or the pendulum bob when you set the system going. (Actually, in both cases you can affect it if you pull it back far enough. In the case of the mass on a spring this is called “breaking” the spring). The Amplitude, on the other hand, is unaffected by the quantities that affect the frequency. You can freely choose the amplitude for any system by how you start the oscillations.
Let’s have a look at the following system, involving two masses attached to each other, and to the walls of a room, by three springs. http://www.npl.co.uk/ssfm/ssfm1/tt/ssfm_interactive_examples/springies/spring_applet.html
This system is much more complicated than a system with one mass bouncing on the end of one spring. If the middle spring (attached to both masses) was not there, this would be a much easier system to analyze. We can change the parameters, so the question is, how can we change the system to make it as if the middle spring wasn’t there. What should we do? • Make the masses of the balls both be small • Make the middle spring have a very small spring constant k • Make the middle spring have a very large spring constant k • Make one mass be very large and the other very small
Correct Answer – B If the middle spring has a very small spring constant, then it will produce only a very weak force for a given stretch or compression. This is almost as if it wasn’t there. It will have little affect on the motion of the two masses as long as it is much weaker than the other two springs. In fact, in this Java applet, you can make the middle spring (or any other part of the system) disappear by setting its spring constant to zero. Check this answer http://www.npl.co.uk/ssfm/ssfm1/tt/ssfm_interactive_examples/springies/spring_applet.html
Now we have two different, and completely separate systems, each consisting of a mass on a single spring. Let’s try to arrange matters so that one of the masses oscillates with a very high frequency, the other with a very low frequency. • If we want to make mass 1 have a high frequency, what must we do? • Make its mass be large and have a large spring constant driving it • Make its mass be small and have a small spring constant driving it • Make its mass be large and have a small spring constant driving it. • Make its mass be small and have a large spring constant driving it.
Correct Answer – D • Recall that the frequency of this kind of system is given by • = (k/m) So if increase k and increase m (as in a) then the two effects cancel each other out. The same thing happens if we decrease both quantities (answer b). We have to increase k and decrease m to make w get bigger. Check your answer http://www.npl.co.uk/ssfm/ssfm1/tt/ssfm_interactive_examples/springies/spring_applet.html
What will happen if we bring back the middle spring and make it quite a stiff spring, that is to say a large spring constant k? • Mass 1 will have an increased frequency, but mass 2 will have a decreased frequency • Mass 1 will have a decreased frequency, but mass 2 will have an increased frequency • Both masses will oscillate with an increased frequency • Both masses will oscillate with a decreased frequency
Correct Answer – C Although the system is now much more complex than we want to analyze mathematically, it still seems reasonable that including another spring with a spring constant a little higher than the two already in place will tend to make each mass oscillate faster. Remember, a stiff spring produces more force, which permits more acceleration (while a large mass resists acceleration). Check your answer http://www.npl.co.uk/ssfm/ssfm1/tt/ssfm_interactive_examples/springies/spring_applet.html
Finally a question purely for fun. • Suppose we make the middle spring have a very very large spring constant. Let’s say a 100 times greater than the other two springs? Which of the following sounds plausible … • The two masses will now be tightly linked together, so they will now tend to oscillate together as if they were only one object • The middle spring will cause the two masses to vibrate in opposite directions with very high frequency • The larger mass will almost stand still while the smaller mass will be a blur • The two balls won’t oscillate at all
Correct Answer – A A very stiff spring resists any kind of stretching or compression very strongly. Since the other two springs are so much weaker (100 times weaker) they will hardly stretch or compress the middle spring, and it will look as if the two masses are tied together and act like one object. Check your answer http://www.npl.co.uk/ssfm/ssfm1/tt/ssfm_interactive_examples/springies/spring_applet.html