Simple Harmonic Motion of a sewing machine

  • #1

Homework Statement


A sewing machine needle moves up and down in simple harmonic motion with an amplitude of 1.27cm and a frequency of 2.55Hz. How long does it take to travel 11.43cm?


Homework Equations


x=A*cos(w*t)
w=2[tex]\pi[/tex]*f
w=2[tex]\pi[/tex]/T
T=1/f


The Attempt at a Solution


I can't determine the proper way to solve for t, since that's what the question is asking for. Every time I try to solve for t, the calculator gives me a 'math error' message and won't let me compute.

The answer is 0.878s, but I need to know how to get it.
 

Answers and Replies

  • #2
179
0
Every time the the needle goes through one full period it travels 4 amplitudes.
 
  • #3
tiny-tim
Science Advisor
Homework Helper
25,832
251
Welcome to PF!

Hi rspbrrylmnd! Welcome to PF! :smile:

(have a pi: π and an omega: ω :wink:)

Show us what you tried, including your wave equation. :smile:
 
  • #4
Ok. I tried first to solve the equation for t before putting in any numeric values. Every time I would do that, though, I'd get the same equation.

x=A*cos(w*t)
x/A=cos(w*t)

From here, I assumed that the next logical step would be to take the inverse cosine of (x/A)

cos[tex]^{-1}[/tex](x/A)=w*t

Then I divided the entire thing by w to get:

[cos[tex]^{-1}[/tex](x/A)]/w=t

This is the same equation I come up with to solve for the time it takes for the needle to travel 11.43 cm. Whenever I plug in the numeric values, though, I can never calculate this equation because 11.43/1.47=9, and one cannot take the inverse cosine of anything larger than 1.

Is there a different relationship between these specific variables that I'm not aware of that would allow me to solve for the time? I'm simply at a loss and/or I have a mental block that is just not allowing me to see this problem from a different angle than by the process I have already tried.
 
  • #5
3,758
425
x is the position and not the distance traveled.
x is always between -1.27 and + 1.27 cm.
You may start by calculating how many full periods (actually full half-periods) are required and then us the formula for the remainder. See post by Jebus_Chris too.
 
  • #6
tiny-tim
Science Advisor
Homework Helper
25,832
251
… Whenever I plug in the numeric values, though, I can never calculate this equation because 11.43/1.47=9, and one cannot take the inverse cosine of anything larger than 1.
x is the position and not the distance traveled.
x is always between -1.27 and + 1.27 cm.
You may start by calculating how many full periods (actually full half-periods) are required and then us the formula for the remainder. See post by Jebus_Chris too.
Every time the the needle goes through one full period it travels 4 amplitudes.
:wink:
 

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