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The tip of a tuning fork goes through 440 complete vibrations in 0.500 s. Find the angular frequency and the period of the motion.

Short Answer

Expert verified
The angular frequency is \( 1760\pi \) rad/s and the period is approximately 0.001136 s.

Step by step solution

01

Understand the Problem

We are given that a tuning fork vibrates 440 times in 0.500 seconds. We need to find the angular frequency and the period of this motion. Angular frequency is related to the frequency of vibration, while the period is the time it takes to complete one full cycle.
02

Calculate Frequency

Frequency is the number of vibrations per second. It is calculated using the formula \( f = \frac{N}{t} \), where \( N \) is the number of vibrations and \( t \) is the time in seconds. Here, \( N = 440 \) and \( t = 0.500 \) seconds.\[ f = \frac{440}{0.500} = 880 \text{ Hz} \]
03

Calculate Angular Frequency

Angular frequency \( \omega \) is related to the frequency \( f \) by the formula \( \omega = 2\pi f \). Thus, we substitute \( f = 880 \text{ Hz} \) into this formula to get:\[ \omega = 2 \pi \times 880 = 1760 \pi \text{ rad/s} \]
04

Calculate the Period

The period \( T \) is the reciprocal of the frequency, calculated as \( T = \frac{1}{f} \). Substituting \( f = 880 \text{ Hz} \):\[ T = \frac{1}{880} \approx 0.001136 \text{ s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Frequency
Angular frequency, often denoted by the symbol \( \omega \), represents how quickly an object rotates in a circular path per unit time. It is a key concept in harmonic motion, where it describes the rate of oscillation. Angular frequency is measured in radians per second (rad/s), which considers the angle that an object covers over a period of time.When dealing with harmonic motion like the vibrations of a tuning fork, we often relate angular frequency to the standard frequency, \( f \), which is measured in hertz (Hz). The relationship is given by the formula:
  • \( \omega = 2\pi f \)
This formula allows us to convert between the more tangible concept of frequency (vibrations per second) to the angular terms, offering a complete picture of the vibration's dynamics.
Vibration Period
The vibration period, noted as \( T \), is the time it takes for one complete cycle of vibration to occur. A cycle is any repeating unit in an oscillation, like the back-and-forth movement of a pendulum.It's crucial to understand that the period is inversely related to frequency. This means the higher the frequency, the shorter the period. The formula for calculating the period \( T \) from frequency \( f \) is:
  • \( T = \frac{1}{f} \)
Knowing the period of a vibration helps us predict the timing of subsequent cycles. For instance, with a high-frequency tuning fork, the period will be very short, indicating rapid oscillations. This immediate understanding is essential for applications like sound synthesis or analyzing the vibrations of different materials.
Frequency Calculation
Frequency calculation is one of the fundamental steps in analyzing harmonic motion. Frequency \( f \) is defined as the number of complete vibrations, or cycles, that occur in a unit of time, typically expressed in hertz (Hz).To calculate frequency in simple scenarios like our tuning fork example, you use the formula:
  • \( f = \frac{N}{t} \)
where \( N \) is the total number of vibrations, and \( t \) is the time in seconds over which these vibrations occur.Understanding how to calculate frequency provides a basis for determining other properties like the angular frequency and the period of motion. Both the precision of measurement and a grasp of these connections are important, as they allow you to transition fluidly between different metrics of vibration and harmonic motion.

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Most popular questions from this chapter

(a) \(\textbf{Music}\). When a person sings, his or her vocal cords vibrate in a repetitive pattern that has the same frequency as the note that is sung. If someone sings the note B flat, which has a frequency of 466 Hz, how much time does it take the person's vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords? (b) \(\textbf{Hearing}\). When sound waves strike the eardrum, this membrane vibrates with the same frequency as the sound. The highest pitch that young humans can hear has a period of 50.0 \(\mu\)s. What are the frequency and angular frequency of the vibrating eardrum for this sound? (c) \(\textbf{Vision}\). When light having vibrations with angular frequency ranging from 2.7 \(\times\) 10\(^{15}\) rad/s to 4.7 \(\times\) 10\(^{15}\) rad/s strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light? (d) \(\textbf{Ultrasound}\). High frequency sound waves (ultrasound) are used to probe the interior of the body, much as x rays do. To detect small objects such as tumors, a frequency of around 5.0 MHz is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

When a body of unknown mass is attached to an ideal spring with force constant 120 N/m, it is found to vibrate with a frequency of 6.00 Hz. Find (a) the period of the motion; (b) the angular frequency; (c) the mass of the body.

A certain simple pendulum has a period on the earth of 1.60 s. What is its period on the surface of Mars, where \(g =\) 3.71 m/s\(^2\)?

A guitar string vibrates at a frequency of 440 Hz. A point at its center moves in SHM with an amplitude of 3.0 mm and a phase angle of zero. (a) Write an equation for the position of the center of the string as a function of time. (b) What are the maximum values of the magnitudes of the velocity and acceleration of the center of the string? (c) The derivative of the acceleration with respect to time is a quantity called the \(jerk\). Write an equation for the jerk of the center of the string as a function of time, and find the maximum value of the magnitude of the jerk.

A harmonic oscillator has angular frequency \(\omega\) and amplitude \(A\). (a) What are the magnitudes of the displacement and velocity when the elastic potential energy is equal to the kinetic energy? (Assume that \(U =\) 0 at equilibrium.) (b) How often does this occur in each cycle? What is the time between occurrences? (c) At an instant when the displacement is equal to \(A\)/2, what fraction of the total energy of the system is kinetic and what fraction is potential?

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