## Particle moving in Simple Harmonic Motion

1. The problem statement, all variables and given/known data

A particle moving in simple harmonic motion with a period T = 1.5 s passes through the equilibrium point at time t0 = 0 with a velocity of 1.00 m/s to the right. A time t later, the particle is observed to move to the left with a velocity of 0.50 m/s. (Note the change in direction of the velocity.) The smallest possible value of the time t is

2. Relevant equations

V= -Aω sin(ωt)

3. The attempt at a solution

If it's moving through eq point at t0=0 with v=1.0m/s then vmax=1.00m/s
vmax=Aω So,

-0.50= -1.00sin (ωt)
sin-1(0.50)= 2pi/T x t
pi/6=2pi/T x t
t= pi/6 x T/2pi = 0.125 sec.

If T=1.5 seconds and it is moving to the right initially then half way through the motion is t=.75 so does that mean that it only begins to move left at t=0.75 seconds after it has reached it max point to the right? Therefore the 0.125 second value I got from the above equation should be added to 0.75 seconds to get the minimum value for t when the particle is moving left at 0.50 m/s?

I feel like I'm missing something. There were no examples of this type of problem done in discussion, not much explanation in lecture, and the book is useless so I feel like I'm going at this pretty blind. Any help explaining this problem to me would be great! Thanks!
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 V= -Aω sin(ωt) ...... If t=0, v=1.0m/s then above equation not valid.
 Yes, I guess that doesn't make sense using the sine. I must have gotten confused along the way as there was just no explanation given for this section except a list of these formulas and I obviously applied it incorrectly. Am I on the right path that Vmax is 1.00 m/s at t=0 if at t=0 the particle passes through the equil. point? If yes, then I think I need to use the acceleration equation instead which involves the cosine. In this case with a=Aw^2 cos (wt) then I get t=.25 seconds. Now my question is, do I add this value to .75 or subtract to get the min. value for t in this question? I would think I would add it, but it appears the answer key for this question indicates the answer is 0.50 seconds which, if I did everything else correctly, indicates that I need to subtract this t value from the T/2 value of .75, but I don't understand why. I would think that if the period involves a movement from the left to the right and then back to the left then halfway through is when the particle is all the way to the right before coming back to the left. This implies to me that for the particle to be going back toward the starting point (to the left) that it would have to be at some t after .75 seconds but before 1.5 seconds. Unless I am to add the .25 value to the T/2 value to get 1.00 seconds and then subtract this from 1.50 to get 0.50 but it still doesn't sit right in my head that the particle would be moving in the opposite direction at a time less than half the period. Any help on the explanation or do I have something wrong all together? Thanks.

## Particle moving in Simple Harmonic Motion

If you use sine function and shifted 90° to the left(leading) so at time =0(max value) it will become cosine function. If you start with cosine function and shifted right it will become sine function.

Is it not possible the velocity function is a cosine function?
 Thank you for the graph. I have to admit I'm a bit unsure of how to work this then, however. I have had zero explanation on how to rework these problems to use an appropriate function (equation) or on aspects like the phase constant, etc. So, now I just feel utterly stupid on these problems. I thought I was beginning to grasp it, but I guess not. Physics is not my strong area and I have to attempt to teach myself in this course due to severe lack of instruction so it's getting frustrating. Is there a really fantastic link someone could direct me to? I'm in algebra based physics if it makes a difference. I have tried endlessly searching online for a good video but have yet to come across something that really explains these different functions, how to use the equations, etc. Thanks so much for any help and your patience.
 Calculus Concepts and Contexts by James Steward is very good book. It starts with Functions and Models. Very valuable information about functions.
 I think you need to explain what course you are studying to get good advice. "Algebra based physics" means nothing to me.

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