 ## ← 07-51 Adjusting the Period

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Showing Revision 1 created 07/23/2012 by Amara Bot.

1. Okay. So here's our mass hanging from the ceiling on a spring with the spring constant K.
2. If I started shaking up and down and oscillating, we know it will move with a certain period.
3. How do we adjust that period? Well let me first tell you the force law of a spring.
4. And to do this, it actually may be better to think of the spring being horizontal.
5. Here's our spring at its normal happy equilibrium position.
6. We're going to call the equilibrium position x = 0.
7. Now let's see, when we stretched the spring a bit and when we moved the mass
8. back to let's say here to some position x.
9. Well the spring exerts a force backwards back towards the equilibrium.
10. The force of a spring is somehow negative. If I pulled to the right, it pulls to the left.
11. And in fact, if I pull further to the right it pulls harder to the left.
12. If you ever pulled on a spring you can tell pulling it a little bit is easy.
13. Pulling it more is more difficult so actually the force is proportional to how much you stretch it.
14. And what's the constant of proportionality? Well, that's why we have this letter K.
15. This is the force on a spring and we know from our good buddy Newton
16. that forces cause accelerations.
17. In this case, the spring will accelerate this way due to the spring force.
18. Why don't we replace this F with this -Kx?
19. Well in the case we get ma = -Kx and so a = -K/m*x.
20. Now I hoped you followed along so far. The acceleration is equal to -K/m*x.
21. The reason why I said I hoped you followed along is because unfortunately now
22. I have to do something that I hate to do.
23. I have to just tell you something and that something is that whenever you can
24. write a force law like this that the acceleration is equal to minus something times x
25. times its displacement from its equilibrium, well it turns out that this something
26. is always equal to ω².
27. And remember omega is our radians per second, our angular frequency.
28. Now let's say you have a spring with a spring constant of 50 N/m.
29. And let's say you want your period to be equal to 1 sec, and remember T = 2π/ω.
30. And why do we want 1 second? Well we're trying to design a clock here.
31. Of course we want a 1-second period. We want it tick tock, tick tock.
32. The question is with this spring and this desired period, what mass do we need?