Question

A weight is suspended from a spring and is moving up and down in a simple harmonic motion . At start , the weight is pushed up 12 cm above the resting position , and then released . After 26 seconds , the weight reaches again to its highest position Find the equation of the motion , and locate the weight with respect to the resting position after 25 seconds since it was released
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Answer 1

The weight is 11.651 centimeters above the resting position.

Procedure - Postion formula for a weight under simple harmonic motion

Simple harmonic motion model

Mathematically speaking, a simple harmonic motion is described by the following formula:

$y = A\cdot \cos \left(\frac{2\pi\cdot t}{T}) + y_{o}$ (1)

Where:

• $t$ - TIme, in seconds.
• $T$ - Period, in seconds.
• $A$ - Amplitude, in centimeters.
• $y_{o}$ - Resting position, in centimeters.

Current position of the weight with respect to the resting position

Please notice that the period is the time needed by the weight to complete one cycle. If we know that $A = 12\,cm$, $y_{o} = 0\,cm$, $T = 26\,s$ and $t = 25\,s$, then the current position of the weight is:

$y = 12\cdot \cos \left(\frac{2\pi\cdot 25}{26} \right)+0$

$y = 11.651\,cm$

The weight is 11.651 centimeters above the resting position. $\blacksquare$

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