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Ch 15: Oscillations
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 15: OscillationsProblema 45b
Capitolo 15, Problema 45b

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

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Identify the key variables given in the problem: the mass of the disk \( m = 0.10 \ \text{g} = 0.0001 \ \text{kg} \), the frequency of oscillation \( f = 1.0 \ \text{MHz} = 1.0 \times 10^6 \ \text{Hz} \), and the amplitude \( A \) (not explicitly given, but it will be used symbolically in the solution).
Recall the formula for the maximum speed in simple harmonic motion (SHM): \( v_{\text{max}} = \omega A \), where \( \omega \) is the angular frequency and \( A \) is the amplitude.
Calculate the angular frequency \( \omega \) using the relationship \( \omega = 2 \pi f \). Substitute \( f = 1.0 \times 10^6 \ \text{Hz} \) into the formula: \( \omega = 2 \pi (1.0 \times 10^6) \).
Substitute the calculated \( \omega \) and the amplitude \( A \) into the formula \( v_{\text{max}} = \omega A \). This will give the maximum speed in terms of the amplitude.
Conclude that the maximum speed depends on the amplitude \( A \), and the final expression for \( v_{\text{max}} \) is \( v_{\text{max}} = (2 \pi f) A \). To find the numerical value, the amplitude \( A \) must be provided.

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Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object moves back and forth around an equilibrium position. In SHM, the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This motion can be described by sinusoidal functions, and key parameters include amplitude, frequency, and maximum speed.
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Maximum Speed in SHM

The maximum speed of an object in Simple Harmonic Motion occurs as it passes through the equilibrium position. It can be calculated using the formula v_max = Aω, where A is the amplitude of the motion and ω is the angular frequency, given by ω = 2πf, with f being the frequency. This relationship highlights how both amplitude and frequency influence the maximum speed.
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Frequency and Angular Frequency

Frequency is the number of cycles of motion that occur in one second, measured in hertz (Hz). Angular frequency, denoted as ω, relates to frequency through the equation ω = 2πf, converting cycles per second into radians per second. Understanding these concepts is crucial for calculating the maximum speed of the transducer in SHM, as they directly affect the motion's characteristics.
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