Simple Harmonic Motion?

Question

The diaphragm of a speaker has a mass of 50.0 g and responds to a signal of frequency 2.0 kHz by moving back and forth with an amplitude of 1.8 X 10^(-4)m at that frequency. (a) what is the max force acting on the diaphragm? (b) what is the mechanical energy of the diaphragm?

Answer 1

OK, first we need a formula for the motion, which, if it's simple harmonic motion is
f(x) = Asin(kx)
where A is 1.8*10^-4
and k = 2000*2*pi
So f(x) = 1.8*10^-4*sin(4000*pi*x)

To compute the maximum force, we need the accelleration f''(x)
f'(x) = 1.8*10^-4*(cos(4000*pi*x))*4000*pi, or
1.8*4*pi*10^-1*cos(4000*pi*x)

f''(x) = 1.8*4*pi/10*(-sin(4000*pi*x))*4000*pi
or
1.8*16*pi^2*100*(-sin(4000*pi*x))
the MAXIMUM acceleration is then 1.8*16*pi^2*100.
This is because you can find a value for x where -sin(4000*pi*x) is 1.

Using F=ma, the maximum force is 50 *1.8*16*pi^2*100 KG m/s^2, or approximately 1.42*10^6 KG m/s^2