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Expert: - 11/19/2011


Question
Hi Scott, I'm really stumped on this question.  Any help in isolating the equation would be really helpful.  The question is:

A weight is attached to a spring and is suspended from a ceiling so that it lies 30 cm from the ceiling when it is at rest.  The spring is stretched so that the weight is 38 cm from the ceiling and is then released.  It bounces up and down for a while.  Each time it comes to the bottom of its bounce, the length of the stretch is 75% of the previous stretch.  The period of each oscillation of the weight is 1/2 a second.  Write an equation that represents the distance from the weight to the ceiling.

(**I tried various types of formula structures to apply to this question, such as the compound interest formula but was not able to get the right bounce when graphing it using a sin function.)

Many thanks for your help!

Answer
30 cm is at rest; stretched down to 38; stretch is reduced b 3/4 each time it bounces.

That sounds like a cos() wave with a minor adjustment.
The cos() wave has an oscillation of 2 (from 1 to -1), and the original oscillation here is 16, since it starts at 8 below and would go to 8 above if there was no resistance.
Also, the period is 1/2 second, and the period of the  cos() curve is 2*pi,
so we would need to divide t by 4*pi.

Since there  is a reduction to 3/4 of what it use to be each time it has passed on oscillation, the curve would be 0.75^[t/(2*pi)]cos[t/(4*pi)], where the angle is in radians.
We would also need a negative involved, since the cos() curve starts in the positive and
the weight starts in the negative.  The last thing we need to do is multiply the curve by 8 since we need it to start at -8 rather than -1.

This gives 8{0.75^[t/(2*pi)]}cos[t/(4*pi)],  

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Scott A Wilson

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

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