Expert: Sherry Wallin - 10/27/2009

Question
A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by the equation below, where y is the distance from equilibrium (in feet) and t is the time (in seconds).
y =1/6sin(3t) + 1/8cos(3t)
(a) Use a graphing utility to graph the model.

(b) Use the identity below, where C = arctan(b/a), a > 0, to write the model in the form y = √a2 + b2sin(Bt + C). Use a graphing utility to verify your result.
a sin(Bθ) + b cos(Bθ) = √a2 + b2sin(Bθ + C)

1.)Y=

2.)Find the amplitude of the oscillations of the weight in feet

3.)Find the frequency of the oscillations of the weight in 1/s

Answer
It is not clear what 1.) is unless it is part (b) and if so then y = 2sqrt(1/6)+(1/4)sin(3t + 37)
where arctan(b/a)= arctan[(1/8)/(1/6)] = arctan (6/8) = arctan(3/4) = 36.87 ~= 37 deg
the amplitude is just the magnitude which is sqrt[(1/6)^2 + (1/8)^2]= sqrt(25/576) = 5/24ft
the frequency is w/2pi where w is the coefficient on the angle so here it is 3 so frequency is 3/2pi

A few notes about typing in math and symbols for clarity:
in your formula for writing your model you wrote √a2 and I think what you mean by that is √a(2) and if so you should put your constant in front of the radical or use parentheses like I did so it is clearer as 2√a, otherwise the way you have it written it is saying take the square root of (2a). Also when typing in fractions use parentheses, the way you have 1/6 and 1/8 written (because it is next to another quantity, standing alone like I just wrote is fine) says something entirely different than what is intended. Saying 1/6sin(3t) is saying 1/(6sin(3t)) and not (1/6)*sin(3t). Notice when I said above 3/2pi I didn't need parentheses because I mean 3 divided by the product of 2 and pi or 3/(2*pi).

Also you have 'Find the frequency of the oscillations of the weight in 1/s' and the 1/s has no meaning, so maybe this is a typo or maybe you want to know the number of oscillations per second??

I hope this has been helpful.

Math Prof