An ultrasonic transducer, of the type used in medical...

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk $(m=0.10 \mathrm{g})$ driven back and forth in SHM at $1.0 \mathrm{MHz}$ by an electromagnetic coil. a. The maximum restoring force that can be applied to the disk without breaking it is $40,000 \mathrm{N}$. What is the maximum oscillation amplitude that won't rupture the disk? b. What is the disk's maximum speed at this amplitude?

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk $(m=0.10 \mathrm{g})$ driven back and forth in SHM at $1.0 \mathrm{MHz}$ by an electromagnetic coil.
a. The maximum restoring force that can be applied to the disk without breaking it is $40,000 \mathrm{N}$. What is the maximum oscillation amplitude that won't rupture the disk?
b. What is the disk's maximum speed at this amplitude?

Key Concepts

Maximum Speed in SHM

The maximum speed in a simple harmonic oscillator occurs when the object passes through its equilibrium position, where all the stored potential energy is converted to kinetic energy. This speed is given by the product of the angular frequency and the oscillation amplitude, and it is a key measure of the dynamic performance of oscillatory systems.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic oscillation where the restoring force is directly proportional to the negative of the displacement from the equilibrium position. This results in sinusoidal oscillations over time, and the behavior is characterized by parameters such as amplitude, frequency, and phase, forming the fundamental basis of many physical systems.

Restoring Force and Amplitude Relationship

In systems undergoing SHM, the maximum restoring force is directly proportional to the amplitude of the motion through an effective force constant. This relationship, rooted in Hooke's Law for elastic systems, is fundamental when determining the safe or maximum displacement a system can experience without failure or structural damage.

Angular Frequency

Angular frequency, defined as 2? times the ordinary frequency, represents how rapidly an oscillator traverses its cycle. It is a critical parameter in SHM because it not only connects the time period of oscillations to the system's dynamics, but also plays an essential role in determining quantities like acceleration and maximum speed.

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