What Is the Maximum Oscillation Amplitude for an Ultrasonic Transducer?

[TJC747]

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk (m = 0.12 g) driven back and forth in SHM at 0.9 MHz by an electromagnetic coil.
(a) The maximum restoring force that can be applied to the disk without breaking it is 36,000 Newtons. What is the maximum oscillation amplitude that won't rupture the disk?
(in µm)

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

For position, I would guess to use x = Acos(omega*t)
I cannot tie the proper equations together, though. Help would be appreciated. Thanks.

 

[ehild]

You know mass and force. What quantity is determined by their ratio?

You wrote correctly the time dependence of position in SHM. What about velocity and acceleration?

ehild

 

[kerimek]

I would approach this question by first understanding the basic principles of SHM (simple harmonic motion) and how it relates to the maximum oscillation amplitude. SHM is a type of motion where the restoring force is directly proportional to the displacement from equilibrium and is always directed towards the equilibrium point. This means that as the amplitude increases, the restoring force also increases.

(a) To determine the maximum oscillation amplitude that won't rupture the disk, we can use the equation for maximum displacement in SHM, which is given by:

A = F_max / (m * omega^2)

Where A is the maximum amplitude, F_max is the maximum restoring force, m is the mass of the disk, and omega is the angular frequency (2*pi*f). Plugging in the given values, we get:

A = (36,000 N) / (0.12 kg * (2*pi*0.9 MHz)^2) = 0.049 µm

Therefore, the maximum oscillation amplitude that won't rupture the disk is 0.049 µm.

(b) To determine the disk's maximum speed at this amplitude, we can use the equation for velocity in SHM, which is given by:

v = A * omega

Where v is the velocity, A is the amplitude, and omega is the angular frequency. Plugging in the values, we get:

v = (0.049 µm) * (2*pi*0.9 MHz) = 0.083 m/s

Therefore, the disk's maximum speed at this amplitude is 0.083 m/s.