# Homework Help: What is the amplitude of sinusoidal motion?

1. Feb 10, 2012

### Aar

1. The problem statement, all variables and given/known data

An object with mass 0.222 kg is hung on a spring whose spring constant is 86.4 N/m. The object is subject to a resistive force given by -bv, where v is its velocity. The ratio of the damped frequency to the undamped (natural) frequency) is 0.886. If this system is subjected to a sinusoidal driving force given by

F(t)=(8.75 N)sin(0.901ωo t) ,

what is the (steady-state) amplitude (in cm) of the resulting sinusoidal motion

2. Relevant equations

A = A$_{o}$/$sqrt{( (1-r^2)^2 + (r^2/q^2) )}$

r = w$_{d}$/w$_{o}$

Q = (w$_{o}$*M)/b

A$_{0}$ = Fm/k

F(t) = (Fm)cos(w$_{d}$*t)

w$_{d}$ = $\sqrt{w_{o}-(b/2m)^2}$

w^2 = k/m

3. The attempt at a solution

this is what I attempted initially and obtained an answer that was correct however in attempting other questions with different numbers the answers which I obtained were not correct. I am wondering if it was coincidental that the initial answer I obtained worked out to be a correct value or if it was the correct method and the program which I submitted the answer in was not registering the answers correctly.

initially solve for w$_{0}$ w$_{d}$ than obtain b

w$_{0}$ = $\sqrt{86.4/0.222}$ = 19.73

w$_{d}$ = w$_{0}$*0.866 = 17.47

b = 2/m * $\sqrt{(19.73)^2 - (17.47)^2}$ = 4.0615

q = 19.73*0.222/4.0615 = 1.0783

A$_{o}$ = 8.75/86.4 = 0.10127

A = 0.10127/$sqrt{( (1-0.866^2)^2 + (0.866^2/1.0783^2) )}$ = 0.1192 m

convert to cm and the answer obtained is = 11.92cm

my uncertainty is wether or not the value in the sinusoidal driving force equation if the 0.901w$_{o}$ has any effect on determining w$_{d}$.

thanks for any help

Last edited: Feb 10, 2012
2. Feb 11, 2012