# SHM - Ultrasonic Transducers

1. Feb 26, 2013

### wayveon

1. The problem statement, all variables and given/known data
An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk (m = 0.14 g) driven back and forth in SHM at 0.8 MHz by an electromagnetic coil.

(a) The maximum restoring force that can be applied to the disk without breaking it is 34,000 N. What is the maximum oscillation amplitude that won't rupture the disk? (in micrometers)

(b) What is the disk's maximum speed at this amplitude? (in m/s)

2. Relevant equations
Kinetic Energy - (1/2)(m)(v^2)
Vmax = (2pi)(f)(A)

3. The attempt at a solution
I wasn't sure whether I should be interpreting this as as spring or not, since we haven't gotten to magnets yet. So i just though, simple harmonic system have constant total energy curves, and interpreted the maximum kinetic energy of the system as being equivalent of the restoring force.

KE = (1/2)(m)(Vmax^2)

34000N = (1/2)(0.00014kg)[[2pi(0.8E6 Hz)(A)]^2]

A = sqrt( [(34000*2)/0.00014kg] / [[2pi(0.8E6 Hz)]^2] ) = 4.384e-3m, which isn't right for a. Oddly, I can see that if i don't do the square root and instead just divide the resulting quantity by two, I get the correct answer. Makes me suspect that the original solution was programmed wrong, simply slip of a ^ to a *, but I doubt myself.
Where am I going wrong with interpreting this problem? I've looked at other solutions, e.g. this one, but I haven't seen some of the other equations before frankly.

Last edited: Feb 26, 2013
2. Feb 26, 2013

### rude man

This makes no sense to me, equating k.e. to restoring force. Instead:

SHM is x = Acos(wt), right?
So compute d2x/dt2 which reveals max. acceleration. Then, Fmax = m*amax which will give you A.