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

b. What is the disk's maximum speed at this amplitude?

Question

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

b. What is the disk's maximum speed at this amplitude?

Accepted Answer

Hey, everyone. So this problem is dealing with simple harmonic motion. Let's see what it's asking us. We have a thin metal plate with a mass of 0.25 g experiencing oscillations. We're asked to determine the maximum speed attained by the plate given a frequency of the oscillations of 1.5 megahertz. We're also told that the plate shows a maximum acceleration of 1.4 times 10 88 m per second squared. Our multiple choice answers here are a 14.9 m per second. B 18.4 m per second, C 15.7 m per second or D 9.42 m per second. So recalling our basic equations of simple harmonic motion is going to be the key to solving this problem. So we can recall that our maximum acceleration is given by the equation A our amplitude multiple multiplied by omega squared where omega is our angular frequency. And then our maximum speed is given by simply a multiplied by omega and then omega in terms of frequency which we are given in the problem is simply two pi F and so combining these and looking at what was given to us. And the problem we have a maximum acceleration and we have a frequency so we can solve for Omega and we are looking for that maximum speed. So we can rewrite a as our maximum acceleration divided by omega squared. And then in turn, our maximum velocity becomes maximum acceleration divided by omega squared and then multiplied by omega. So one of the Omegas cancel and then we can plug in that two pi F. So we have a max divided by two if and from there, we can plug in the values from the problem. Our maximum acceleration is 1.4 times 10 to the eight meters per second squared divided by two pi multiplied by our frequency which is 1.5. Be careful with your unit. That's megahertz. So 1.5 times 10 to the six Hertz. And so when we plug that in to our populated, we get a maximum speed of 14.9 m per second. And that aligns with answer choice. A so that is the correct answer for this problem. That's all we have for this one. We'll see you in the next video.

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk (m = 0.10 g) driven back and forth in SHM at 1.0 MHz by an electromagnetic coil. b. What is the disk's maximum speed at this amplitude?

Verified Solution

Key Concepts

Simple Harmonic Motion (SHM)

Maximum Speed in SHM

Frequency and Angular Frequency