11/15/2023 0 Comments Simple harmonic motion![]() ![]() ![]() Sometimes, they may not tell you what κ is. You can find θ, you can derivatives and so on. You just stick it in and mindlessly calculate all the formulas. Now, if someone tells you this is κ, then you are done. You can call it A cos (ωt- φ, and ω now will be the ratio of this κ at the moment of inertia, because mathematically that’s the role played by κ and I. Mathematically, that equation’s identical to that, and θ then will look like some constant. If you find this κ then you can say -κ times θ is I times d 2θ over dt 2. If you make θ negative it’ll try to bring you back. That means if you make θ positive, the torque will try to twist you the other way. It’ll still be proportional to θ, and you put a minus sign for the same reason you put a minus sign here to tell you it’s a restoring torque. If θ is not 0, it’ll begin as some function of θ, but the leading term would be just θ. What can be the expression for the restoring torque? When you don’t do anything, it doesn’t do anything, so it’s a function of θ that vanishes when θ is 0. So, now we don’t have a restoring force but we have a restoring torque. But if you come and twist it by angle θ, give it a little twist, then it will try to untwist itself. Now, we don’t have a mass but we have a bar, let’s say, suspended by a cable, hanging from the ceiling, and it’s happy the way it is. But it is a very generic situation, so I’ll give you a second example. This is an example of simple harmonic motion. That means when the time is 0, x will have the biggest value A. That tells you what your clock is doing when x is a maximum, and we can always choose φ to be 0. A is the amplitude, φ is called the phase. The most general answer looks like A cos ωt - φ. And what’s the solution to this equation? I think we did a lot of talking and said look, we are looking for a function which, when differentiated twice, looks like the same function up to some constants, and we know they are trigonometry functions. But if you pull it so the mass comes here, you move it from x = 0 to a new location x then, there is a restoring force F, which is - kx, and that force will be equal to mass times acceleration by Newton’s law, and so you’re trying to solve the equation d 2x/dt 2 = -k/m times x, and we use the symbol, ω 2 = k/m, or ω square root of k/ m. This spring has a certain natural length, which I’m showing you here. The standard textbook example is this mass on spring. Professor Ramamurti Shankar: Stable equilibrium, and if you disturb them, they rock back and forth and there are two simple examples. Example Equations of Oscillating Objects Fundamentals of Physics I PHYS 200 - Lecture 17 - Simple Harmonic MotionĬhapter 1. So we have approximate simple harmonic motion, where w 2 = g/l. If the particle is at P or Q when t = 0, then the following equation also holds:Ī simple pendulum consists of a particle P of mass m, suspended from a fixed point by a light inextensible string of length a, as shown here: If the particle is at 0 when t = 0, then the following equation also holds: The periodof the motion is the time it takes for the particle to perform one complete cycle. Hence the maximum velocity is a w (put x = 0 in the above equation and take the square root). Where v is the velocity of the particle, a is the amplitude and x is the distance from O.įrom this equation, we can see that the velocity is maximised when x = 0, since v 2 = w 2a 2 - w 2x 2 We can solve this differential equation to deduce that: If so, you simply must show that the particle satisfies the above equation. You may be asked to prove that a particle moves with simple harmonic motion. Where w is a constant (note that this just says that the acceleration of the particle is proportional to the distance from O). The amplitudeof the motion is the distance from O to either P or Q (the distances are the same).Ī particle which moves under simple harmonic motion will have the equation The particle will therefore move between two fixed points (P and Q). It will keep going and then again slow down as it reaches P before stopping at P and returning to O once more. ![]() Simple Harmonic Motion arises when we consider the motion of a particle whose acceleration points towards a fixed point O and is proportional to the distance of the particle from O (so the acceleration increases as the distance from the fixed point increases).Īs the particle moves away from the fixed point O, since the acceleration is pointing towards O, the particle will slow down and eventually stop (at Q), before returning to O. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |