/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 The tip of a tuning fork goes th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 3

The tip of a tuning fork goes through 440 complete vibrations in \(0.500 \mathrm{~s}\). Find the angular frequency and the period of the motion.

Short Answer

Expert verified
The angular frequency is approximately \(5536\) rad/s, and the period of the motion is approximately \(0.00114\) s

Step by step solution

01

Calculate Frequency

Since frequency is defined as the number of complete vibrations per unit time, it can be calculated as \(f = \frac{440}{0.5}= 880\) Hz
02

Find Angular Frequency

Angular frequency can be found by the formula \(\omega = 2\pi f\). Substituting \(f = 880\) Hz, we get \(\omega = 2\pi * 880 \approx 5536\) rad/s
03

Find Period

The period of the motion is given by the inverse of the frequency. Thus \(T = 1/f = 1/880 \approx 0.00114\) s

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Frequency
Frequency is a fundamental concept in understanding wave and oscillatory motion. It tells us how often a repeating event occurs in one second. For example, in the case of a tuning fork, frequency tells us how many times the fork vibrates in each second. This is measured in Hertz (Hz), where one Hertz equals one complete cycle or vibration per second.

In our exercise, to find the frequency, we counted the number of vibrations and divided by the time they took. Here, the tuning fork completed 440 vibrations in 0.5 seconds. Using the formula:
  • Frequency, \( f = \frac{\text{Number of Vibrations}}{\text{Time}} \)
Plugging in the numbers:
  • \( f = \frac{440}{0.5} = 880 \) Hz
This frequency value shows that the tuning fork completes 880 cycles every second!
Period
The period of motion is the time it takes for one complete cycle or vibration to occur. It is closely related to frequency, because while frequency counts cycles per second, the period measures the time for each cycle.

Mathematically, period \( T \) is the reciprocal of the frequency:
  • Period, \( T = \frac{1}{f} \)
Understanding the period helps us determine the timing of oscillations. In this exercise, with a frequency of 880 Hz, the period can be calculated as:
  • \( T = \frac{1}{880} \approx 0.00114 \) seconds
This indicates each individual vibration of the tuning fork lasts approximately 0.00114 seconds!
Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic motion where an object moves back and forth along a path in a regular, repeated manner. It is characterized by its sinusoidal oscillations, meaning it follows a sine or cosine wave pattern.

SHM is fundamental in physics because it models how many systems behave naturally, like springs, pendulums, and, as in our exercise, a vibrating tuning fork. Two key properties define SHM:
  • Amplitude: the maximum extent of the vibration or displacement from the equilibrium position.
  • Angular Frequency (\( \omega \)): measures how quickly an object moves through its oscillations and is given by \( \omega = 2\pi f \).
In the exercise example, we calculated the angular frequency as:
  • \( \omega = 2\pi \times 880 \approx 5536 \) rad/s
This angular frequency tells us the rate of the phase change of the tuning fork's vibration. SHM and its mathematical framework help us predict the motion patterns of oscillating systems in everyday life.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Don't Miss the Boat. While on a visit to Minnesota ("Land of 10,000 Lakes"), you sign up to take an excursion around one of the larger lakes. When you go to the dock where the \(1500 \mathrm{~kg}\) boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude \(20 \mathrm{~cm}\) The boat takes \(3.5 \mathrm{~s}\) to make one complete up-and-down cycle. When the boat is at its highest point, its deck is at the same height as the stationary dock. As you watch the boat bob up and down, you (mass \(60 \mathrm{~kg}\) ) begin to feel a bit woozy, due in part to the previous night's dinner of lutefisk. As a result, you refuse to board the boat unless the level of the boat's deck is within \(10 \mathrm{~cm}\) of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion?

BIO Weighing a Virus. In February 2004, scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just \(30 \mathrm{nm}\) long) with a laser, first without the virus and then after the virus had attached itself to the silicon. The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached \(\left(f_{\mathrm{S}+\mathrm{V}}\right)\) to the frequency without the virus \(\left(f_{\mathrm{S}}\right)\) is given by \(f_{\mathrm{S}+\mathrm{V}} / f_{\mathrm{S}}=1 / \sqrt{1+\left(m_{\mathrm{V}} / m_{\mathrm{S}}\right)},\) where \(m_{\mathrm{V}}\) is the mass of the virus and \(m_{\mathrm{S}}\) is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of \(2.10 \times 10^{-16} \mathrm{~g}\) and a frequency of \(2.00 \times 10^{15} \mathrm{~Hz}\) without the virus and \(2.87 \times 10^{14} \mathrm{~Hz}\) with the virus. What is the mass of the virus, in grams and in femtograms?

A sinusoidally varying driving force is applied to a damped harmonic oscillator of force constant \(k\) and mass \(m .\) If the damping constant has a value \(b_{1},\) the amplitude is \(A_{1}\) when the driving angular frequency equals \(\sqrt{k / m}\). In terms of \(A_{1}\), what is the amplitude for the same driving frequency and the same driving force amplitude \(F_{\max },\) if the damping constant is (a) \(3 b_{1}\) and (b) \(b_{1} / 2 ?\)

Four passengers with combined mass \(250 \mathrm{~kg}\) compress the springs of a car with worn-out shock absorbers by \(4.00 \mathrm{~cm}\) when they get in. Model the car and passengers as a single object on a single ideal spring. If the loaded car has a period of vibration of \(1.92 \mathrm{~s}\), what is the period of vibration of the empty car?

A \(40.0 \mathrm{~N}\) force stretches a vertical spring \(0.250 \mathrm{~m}\). (a) What mass must be suspended from the spring so that the system will oscillate with a period of \(1.00 \mathrm{~s} ?\) (b) If the amplitude of the motion is \(0.050 \mathrm{~m}\) and the period is that specified in part (a), where is the object and in what direction is it moving \(0.35 \mathrm{~s}\) after it has passed the equilibrium position, moving downward? (c) What force (magnitude and direction) does the spring exert on the object when it is \(0.030 \mathrm{~m}\) below the equilibrium position, moving upward?
See all solutions

Recommended explanations on Physics Textbooks

Electrostatics

Read Explanation

Medical Physics

Read Explanation

Electricity

Read Explanation

Physical Quantities And Units

Read Explanation

Space Physics

Read Explanation

Turning Points in Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.