Question

A medical ultrasound imaging machine has an ultrasonic transducer (m=0.10 g) which is driven in simple harmonic motion at 1.3 MHz.

A. If the maximum restoring force the disk can withstand is 38,000 N, what is the maximum amplitude that will not destroy the disk.

B. What is the maximum speed of the disk?

Answer 1

We have given the mass, m = 0.10 g

Frequency = 1.3MHz.

${\omega}^2 = \dfrac{k}{m}$

${\omega}^2 m = k$ = ${(2\pi \nu) }^2 m$

So we can determine the value of k (the force constant)

K= ${(2\pi \nu) }^2 m$ = ${(2\pi \times 1.3\times 10^6) }^2 \times 0.10\times 10^{-3} \approx 6.66\times 10^9 N/m$

A)  F= KA where A is maximum amplitude

A = F/K =   $\dfrac{38000}{6.66\times 10^9}\approx 5.71\times 10^{-6}m$

B. ) Max speed V = $A\omega= A(2\pi\nu)$ =  $5.71\times10^{-6}\times (2\pi\times 1.3\times 10^6)$

= 46.64m/s  answer.

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