Solving for t in SHM (complex solution)

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The discussion revolves around solving for the time variable t in the context of simple harmonic motion (SHM) using the equation x(t) = -2/3cos10t + 1/2sin10t. The user initially struggles to find when x(t) equals zero and attempts to rearrange the equation to isolate t. After some algebra, they realize their initial solution only provided one instance of t, while the problem requires multiple solutions due to the periodic nature of SHM. Ultimately, they correct their approach by incorporating the period of the motion, leading to a complete solution. The thread highlights the importance of considering all possible solutions in periodic functions.
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Hey guys, this is my first post, I was hoping you all could offer some advice. I'm facing a problem involving free undamped SHM. Everything is working out well so far but I ran into a problem when trying to find t. Here's what I have:

x(t) = -2/3cos10t + 1/2sin10t

Now, if I want to find the times when x(t) = 0... how would I go about that? In a similar problem that only involved a single term, I was able to use cos^-1 to solve for t, but in this case would that still hold? Setting x(t) to 0 I get:

0 = -2/3cos10t + 1/2sin10t

But now I'm stuck.

Any help would be greatly appreciated!
 
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Hi mouser! Welcome to PF! :smile:

mouser said:
0 = -2/3cos10t + 1/2sin10t

Hint: how might you re-arrange this equation? :smile:

[size=-2](if you're happy, don't forget to mark thread "solved"!)[/size]​
 
Thanks!

If I rearrange it to be:

2/3cos10t = 1/2sin10t

4/3cot10t = 1

3/4tan10t = 1

tan10t = 4/3

t = (tan^-1(4/3))/10

The answer is incorrect. Did I make a mistake in that algebra? Thank you for your help!
 
mouser said:
t = (tan^-1(4/3))/10

Looks ok to me!

Except you only have one solution … there should be infinitely many … what are the others? :smile:

(What were you actually asked for? Was it the times, or the period?)
 
The question was "at what time does the mass pass through the equilibrium position heading downward for the second time?"

I just figured out where I went wrong. The answer I got previously was only partially correct. For the correct answer I added the value for one full period and it checks out! eureka!

Thanks tiny-tim!
 
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