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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

An object with height \(h\), mass \(M\), and a uniform cross-sectional area \(A\) floats upright in a liquid with density \(\rho\). (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force with magnitude \(F\) is applied to the top of the object. At the new equilibrium position, how much farther below the surface of the liquid is the bottom of the object than it was in part (a)? (Assume that some of the object remains above the surface of the liquid.) (c) Your result in part (b) shows that if the force is suddenly removed, the object will oscillate up and down in SHM. Calculate the period of this motion in terms of the density \(\rho\) of the liquid, the mass \(M\), and the cross- sectional area A of the object. You can ignore the damping due to fluid friction (see Section 14.7).

A rifle bullet with mass 8.00 g and initial horizontal velocity 280 m/s strikes and embeds itself in a block with mass 0.992 kg that rests on a frictionless surface and is attached to one end of an ideal spring. The other end of the spring is attached to the wall. The impact compresses the spring a maximum distance of 15.0 cm. After the impact, the block moves in SHM. Calculate the period of this motion.

A holiday ornament in the shape of a hollow sphere with mass \(M =\) 0.015 kg and radius \(R =\) 0.050 m is hung from a tree limb by a small loop of wire attached to the surface of the sphere. If the ornament is displaced a small distance and released, it swings back and forth as a physical pendulum with negligible friction. Calculate its period. (\(Hint\): Use the parallel-axis theorem to find the moment of inertia of the sphere about the pivot at the tree limb.)

You hang various masses \(m\) from the end of a vertical, 0.250-kg spring that obeys Hooke's law and is tapered, which means the diameter changes along the length of the spring. Since the mass of the spring is not negligible, you must replace \(m\) in the equation \(T =\) 2\(\pi\sqrt{ m/k }\) with \(m + m_\mathrm{eff}\), where \(m_\mathrm{eff}\) is the effective mass of the oscillating spring. (See Challenge Problem 14.93.) You vary the mass m and measure the time for 10 complete oscillations, obtaining these data: (a) Graph the square of the period \(T\) versus the mass suspended from the spring, and find the straight line of best fit. (b) From the slope of that line, determine the force constant of the spring. (c) From the vertical intercept of the line, determine the spring's effective mass. (d) What fraction is \(m_\mathrm{eff}\) of the spring's mass? (e) If a 0.450-kg mass oscillates on the end of the spring, find its period, frequency, and angular frequency.

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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