Ultrasonic transducer pretty much another SHM problem

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SUMMARY

The discussion centers on calculating the maximum oscillation amplitude of an ultrasonic transducer, specifically a thin disk with a mass of 0.10 g, driven in simple harmonic motion (SHM) at a frequency of 1.0 MHz. The maximum restoring force before rupture is 40,000 N. The correct calculation for the spring constant (k) is derived using the formula k = m(2πf)², resulting in k = 628 N/m. The maximum amplitude that does not rupture the disk is determined to be 10.1 micrometers, correcting the initial miscalculation of 63.66 m.

PREREQUISITES
  • Understanding of simple harmonic motion (SHM)
  • Familiarity with the spring constant (k) and its calculation
  • Basic knowledge of force equations, specifically Fsp = -kx
  • Ability to convert units, particularly between meters and micrometers
NEXT STEPS
  • Study the principles of simple harmonic motion in greater detail
  • Learn about the applications of ultrasonic transducers in medical imaging
  • Explore advanced calculations involving spring constants and oscillation amplitudes
  • Investigate the effects of material properties on the performance of ultrasonic devices
USEFUL FOR

Students in physics or engineering, particularly those focusing on acoustics, mechanical vibrations, or medical imaging technologies.

<3Science
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Homework Statement



An ultrasonic transducer, of the type used medical ultrasound imaging, is avery thin disk (m=0.10 g) driven back and forth in SHM at 1.0 MHz by an electromagnetic coil.

a. The maximum restoring force that can be applied to the disk without breaking it is 40,000 N. What is the maximum oscillation amplitude that won't rupture the disk?


Homework Equations



f = 1/2pi * sqrt (k/m)
Fsp = -kx

The Attempt at a Solution



I figured I could get k so I went

k = m(2(pi)f) ^2
k = (0.00010 kg) (2pi * 1.0*10^6 Hz) = 628

And then put it into the second equation to get max amplitude

40,000 N = 628 x

x = 63.66 ... m ? It's however not correct the answer is supposed to be 10.1 micrometers
 
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welcome to pf!

hi <3Science! welcome to pf! :smile:

(have a pi: π and try using the X2 icon just above the Reply box :wink:)
<3Science said:
k = m(2(pi)f) ^2
k = (0.00010 kg) (2pi * 1.0*10^6 Hz) = 628

erm :rolleyes: … squared? :redface:
 

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