Simple Harmonic Motion of a sewing machine

AI Thread Summary
A sewing machine needle exhibits simple harmonic motion with an amplitude of 1.27 cm and a frequency of 2.55 Hz. The challenge is to determine the time it takes for the needle to travel 11.43 cm. The correct approach involves recognizing that the needle's position, x, is limited to the amplitude range, meaning it cannot exceed ±1.27 cm. To find the total travel time, one must calculate the number of full periods required for the distance and then account for any additional motion. The solution ultimately reveals that the time taken is 0.878 seconds.
rspbrrylmnd
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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\pi*f
w=2\pi/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.
 
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Every time the the needle goes through one full period it travels 4 amplitudes.
 
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:
 
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^{-1}(x/A)=w*t

Then I divided the entire thing by w to get:

[cos^{-1}(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.
 
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.
 
rspbrrylmnd said:
… 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.
nasu said:
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.
Jebus_Chris said:
Every time the the needle goes through one full period it travels 4 amplitudes.

:wink:
 
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