# Simple Harmonic Motion of a sewing machine

## 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.

## Answers and Replies

Related Introductory Physics Homework Help News on Phys.org
Every time the the needle goes through one full period it travels 4 amplitudes.

tiny-tim
Science Advisor
Homework Helper
Welcome to PF!

Hi rspbrrylmnd! Welcome to PF!

(have a pi: π and an omega: ω )

Show us what you tried, including your wave equation.

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.

tiny-tim
Science Advisor
Homework Helper
… 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.