/*! 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 6 A small bead of mass \(4.00 \mat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 6

A small bead of mass \(4.00 \mathrm{~g}\) is attached to a horizontal string. Transverse waves of amplitude \(A=0.800 \mathrm{~cm}\) and frequency \(f=20.0 \mathrm{~Hz}\) are set up on the string. Assume the mass of the bead is small enough that the bead doesn't alter the wave motion. During the wave motion, what is the maximum vertical force that the string exerts on the bead?

Short Answer

Expert verified
The maximum vertical force that the string exerts on the bead is \(1.6蟺虏 N\).

Step by step solution

01

Calculate Angular Frequency

The angular frequency 蠅 can be calculated by the formula \(蠅 = 2蟺f\). Here \(f = 20.0 Hz\). Substituting these values, we get \(蠅 = 2蟺 \cdot 20.0 = 40蟺 rad/s\).
02

Calculate Maximum Acceleration

The maximum acceleration \(a_{max}\) can be calculated by the formula \(a_{max} = 蠅^2 \cdot A\). The amplitude \(A = 0.800 cm = 0.008 m\). Substituting these values, we get \(a_{max} = (40蟺)虏 \cdot 0.008 = 400蟺虏 m/s虏\).
03

Calculate Maximum Force

The maximum force \(F_{max}\) can be calculated by the formula \(F_{max} = m \cdot a_{max}\). The mass \(m = 4.00g = 0.004 kg\). Substituting these values we get \(F_{max} = 0.004 \cdot 400蟺虏 = 1.6蟺虏 N\).

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.

Angular Frequency
Angular frequency is a crucial concept when analyzing oscillatory systems, particularly in physics. It specifies how fast the wave oscillates in radians per second. This is different from regular frequency, which is measured in cycles per second or Hertz (Hz). To calculate angular frequency, you use the formula
  • \( \omega = 2\pi f \)
where \( f \) is the frequency.
For our context, if you have a wave with a frequency of 20 Hz, you multiply it by \( 2\pi \) to convert it into radians per second.
This becomes \( \omega = 40\pi \text{ rad/s} \), allowing you to determine other aspects of the wave, like acceleration and forces acting on objects involved in the motion.
Transverse Waves
Transverse waves are a type of wave where the movement of the medium is perpendicular to the direction of the wave's motion. That means if the wave moves horizontally, the medium (like a string or water) moves up and down.
This characteristic of transverse waves is crucial in many areas, including light and electromagnetic waves, which are transverse in nature.
  • Key features to remember include:
  • The medium oscillates at right angles to the direction of energy transfer.
  • Amplitude is the maximum displacement from the rest position.
  • Wavelength is the distance between consecutive crests or troughs.
These features are essential when calculating the forces and motion in systems like the horizontal string supporting a mass.
Maximum Acceleration
Maximum acceleration in wave motion indicates how quickly the speed of a point on the wave can change. It is valuable for understanding both the dynamic forces involved and how they might affect objects in or on the wave.
To find this, we use the formula:
  • \( a_{\text{max}} = \omega^2 \times A \)
where \( \omega \) is the angular frequency and \( A \) is the amplitude.
Using the earlier calculated \( \omega = 40\pi \) rad/s and the amplitude \( A = 0.008 \) m:
  • The maximum acceleration becomes \( a_{\text{max}} = (40\pi)^2 \times 0.008 = 400\pi^2 \) m/s虏.
This value helps determine the maximum vertical force exerted on an object at the wave's peak acceleration.
Vertical Force Calculation
When considering waves, it's essential to calculate the force acting on objects involved. For vertical force calculation, especially with an object attached to a wave like our bead, you consider the mass and the system's maximum acceleration.
The force is determined using Newton鈥檚 Second Law, expressed as:
  • \( F_{\text{max}} = m \times a_{\text{max}} \)
where \( m \) is the mass and \( a_{\text{max}} \) is maximum acceleration.
In this exercise, the given mass is 4 grams or 0.004 kg, and the maximum acceleration was found as \( 400\pi^2 \) m/s虏:
  • Thus, \( F_{\text{max}} = 0.004 \times 400\pi^2 = 1.6\pi^2 \) N.
This force reflects how strongly the wave could potentially impact the bead at its peak.

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

A \(1.50-\mathrm{m}\) -long rope is stretched between two supports with a tension that makes the speed of transverse waves \(62.0 \mathrm{~m} / \mathrm{s}\). What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

(a) A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed \(v,\) frequency \(f,\) amplitude \(A,\) and wavelength \(\lambda .\) Calculate the maximum transverse velocity and maximum transverse acceleration of points located at (i) \(x=\lambda / 2\), (ii) \(x=\lambda / 4,\) and (iii) \(x=\lambda / 8,\) from the left-hand end of the string. (b) At each of the points in part (a), what is the amplitude of the motion? (c) At each of the points in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement?

For a violin, estimate the length of the portions of the strings that are free to vibrate. (a) The frequency of the note played by the open E5 string vibrating in its fundamental standing wave is 659 Hz. Use your estimate of the length to calculate the wave speed for the transverse waves on the string. (b) The vibrating string produces sound waves in air with the same frequency as that of the string. Use \(344 \mathrm{~m} / \mathrm{s}\) for the speed of sound in air and calculate the wavelength of the E5 note in air. Which is larger: the wavelength on the string or the wavelength in air? (c) Repeat parts (a) and (b) for a bass viol, which is typically played by a person standing up. Start your calculation by estimating the length of the bass viol string that is free to vibrate. The G2 string produces a note with frequency \(98 \mathrm{~Hz}\) when vibrating in its fundamental standing wave.

A horizontal wire is tied to supports at each end and vibrates in its second- overtone standing wave. The tension in the wire is \(5.00 \mathrm{~N}\), and the node-to-node distance in the standing wave is \(6.28 \mathrm{~cm}\). (a) What is the length of the wire? (b) A point at an antinode of the standing wave on the wire travels from its maximum upward displacement to its maximum downward displacement in \(8.40 \mathrm{~ms}\). What is the wire's mass?

One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at \(120 \mathrm{~Hz}\). The other end passes over a pulley and supports a \(1.50 \mathrm{~kg}\) mass. The linear mass density of the rope is \(0.0480 \mathrm{~kg} / \mathrm{m} .\) (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to \(3.00 \mathrm{~kg} ?\)
See all solutions

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Read Explanation

Thermodynamics

Read Explanation

Oscillations

Read Explanation

Kinematics Physics

Read Explanation

Particle Model of Matter

Read Explanation

Electrostatics

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.