Notes: An air conditioning unit is vibrating in simple harmonic motion with a period of 0.27 s and a range (from the maximum in one direction to the...

Thursday, 31 July 2014

An air conditioning unit is vibrating in simple harmonic motion with a period of 0.27 s and a range (from the maximum in one direction to the...

Hello!

A simple harmonic motion has a form

$\displaystyle{x}{\left({t}\right)}={A}\cdot{\sin{{\left({b}\cdot{\left({x}-{s}\right)}\right)}}}+{I},$

where $\displaystyle{t}$ is a time, $\displaystyle{x}$ is a position, $\displaystyle{A}$ is an amplitude, $\displaystyle{b}$ is a frequency, $\displaystyle{s}$ is a phase shift and $\displaystyle{I}$ is an initial position.

We may assume $\displaystyle{I}={0}.$ Also, because "at $\displaystyle{t}={0}$ it is at its central position" the phase sift $\displaystyle{s}$ is also zero.

From the minimum to the maximum there are two amplitudes, therefore $\displaystyle{A}={1.5}{c}{m}.$

A frequency is always $\displaystyle{2}\cdot\pi$ ...

Hello!

A simple harmonic motion has a form

$\displaystyle{x}{\left({t}\right)}={A}\cdot{\sin{{\left({b}\cdot{\left({x}-{s}\right)}\right)}}}+{I},$

We may assume $\displaystyle{I}={0}.$ Also, because "at $\displaystyle{t}={0}$ it is at its central position" the phase sift $\displaystyle{s}$ is also zero.

From the minimum to the maximum there are two amplitudes, therefore $\displaystyle{A}={1.5}{c}{m}.$

A frequency is always $\displaystyle{2}\cdot\pi$ divided by a period, therefore $\displaystyle{b}=\frac{{{2}\pi}}{{0.27}}$ $\displaystyle{{s}}^{{-{1}}}.$

So in the given case the position is

$\displaystyle{x}{\left({t}\right)}={1.5}\cdot{\sin{{\left({t}\cdot\frac{{{2}\pi}}{{0.27}}\right)}}}.$

And yes, at $\displaystyle{t}={0}$ it is moving in the positive direction. The expression under the sinus must be in radians, not degrees.

The position for $\displaystyle{t}_{{1}}={55}{s}$ is

$\displaystyle{x}_{{1}}={x}{\left({t}_{{1}}\right)}={1.5}\cdot{\sin{{\left({2}\pi\cdot{\left(\frac{{55}}{{0.27}}\right)}\right)}}}\approx-{1.44}{\left({c}{m}\right)}.$