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 maximum in the other) of 3.0 cm. At t = 0 it is at its central position and moving in the +x direction. What is its position when t = 55 s?

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A simple harmonic motion has a form

`x(t)=A*sin(b*(x-s))+I,`

where `t` is a time, `x` is a position, `A` is an amplitude, `b` is a frequency, `s` is a phase shift and `I` is an initial position.

We may assume `I=0.`  Also, because "at `t=0` it is at its central...

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Hello!

A simple harmonic motion has a form

`x(t)=A*sin(b*(x-s))+I,`

where `t` is a time, `x` is a position, `A` is an amplitude, `b` is a frequency, `s` is a phase shift and `I` is an initial position.

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

From the minimum to the maximum there are two amplitudes, therefore `A=1.5cm.`

A frequency is always `2*pi` divided by a period, therefore `b=(2pi)/0.27` `s^(-1).`

So in the given case the position is

`x(t)=1.5*sin(t*(2pi)/0.27).`

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

 

The position for `t_1=55s` is` `

`x_1=x(t_1)=1.5*sin(2pi*(55/0.27)) approx-1.44 (cm).`

This means 1.44 cm in the negative direction. This is the answer.

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