Ch 15: Oscillations

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 15: Oscillations

Problem 45b

Chapter 15, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

Your lab instructor has asked you to measure a spring constant using a dynamic method—letting it oscillate—rather than a static method of stretching it. You and your lab partner suspend the spring from a hook, hang different masses on the lower end, and start them oscillating. One of you uses a meter stick to measure the amplitude, the other uses a stopwatch to time 10 oscillations. Your data are as follows: Use the best-fit line of an appropriate graph to determine the spring constant.

Textbook Question

Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The spring's length as a function of time is shown in FIGURE P15.46. What is her speed when the spring's length is 1.2 m?

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Textbook Question

A 100 g block attached to a spring with spring constant 2.5 N/m oscillates horizontally on a frictionless table. Its velocity is 20 c/m when 𝓍 = -5.0 cm What is the block's position when the acceleration is maximum?

Textbook Question

Two 500 g air-track gliders are each connected by identical springs with spring constant 25 N/m to the ends of the air track. The gliders are connected to each other by a spring with spring constant 2.0 N/m. One glider is pulled 8.0 cm to the side and released while the other is at rest at its equilibrium position. How long will it take until the glider that was initially at rest has all the motion while the first glider is at rest?

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Textbook Question

When the displacement of a mass on a spring is (½)A, what fraction of the energy is kinetic energy and what fraction is potential energy?

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Textbook Question

A 200 g block hangs from a spring with spring constant 10 N/m. At t = 0 s the block is 20 cm below the equilibrium point and moving upward with a speed of 100 cm/s. What are the block's a. Oscillation frequency?