Decay Factor Form Maths

Question

For my maths assignment, i had to do an experiment using a pendulum and measuring the amplitude of every 10 swings up to the 80th swing.

One of the questions is find the ratio between consecutive columns and average them.

Then it says, 'Given that this is an example of exponential decay of the form; y=a x b^x (where a is the initial value and b is the decay factor), find a formula that models the amplitude as a function of the number of 10-swing periods.'

Please help me find the decay factor and then the question.

Thank you!

 

Data:

Number of Swings            0        10         20      30     40    50     60     70    80

Amplitude of Swing (cm) 162.5  135.0  109.5  89.70  77.0  57.0  48.0  43.0  38.0

Answer 1

y = a* b^x

 

take the logarithm of both sides. Any base will do. I will use the natural log.

 

ln(y) = ln(a) + xln(b). If we set Y=ln(y), A = ln(a) and B = ln(b), we have the equation of a straight line

 

Y = A + Bx.

 

You have a set of data where the number of swings is the x -coordinate and amplitude is the y coordinate. Take the logarithm of all the y's to get the Y's . So now you have coordinates for (x,Y). Now use Google Sheets or Excel to do a straight line fit to the data(x,Y). The slope of the straight line gives you B and the Y intercept gives you A. Then undo the logarithm by forming a = exp(A) and b = exp(B) to get a and b.

 

Hope this helps