Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 15: Problem 34

A string with a mass of \(30.0 \mathrm{~g}\) and a length of \(2.00 \mathrm{~m}\) is stretched under a tension of \(70.0 \mathrm{~N}\). How much power must be supplied to the string to generate a traveling wave that has a frequency of \(50.0 \mathrm{~Hz}\) and an amplitude of \(4.00 \mathrm{~cm} ?\)

Short Answer

Expert verified
Question: Calculate the power needed to generate a traveling wave with a frequency of 50.0 Hz and an amplitude of 4.00 cm on a string with a mass of 30.0 g, a length of 2.00 m, and a tension of 70.0 N. Answer: The power required to generate the traveling wave is approximately 50.58 W.

Step by step solution

01

Calculate the string's linear mass density

To find the wave speed, we first need to calculate the linear mass density (µ) of the string. Linear mass density is mass per unit length. The formula for linear mass density is: \(µ = \frac{mass}{length}\) Given, mass \(m = 30.0 g = 0.030 kg\) and length \(L = 2.00 m\). Now, calculate the linear mass density: \(µ = \frac{0.030 kg}{2.00 m} = 0.015 kg/m\)
02

Calculate the wave speed

The wave speed (v) can be calculated using the formula: \(v = \sqrt{\frac{Tension}{Linear~mass~density}}\) With Tension \(T = 70.0 N\), and the calculated linear mass density \(µ = 0.015 kg/m\). Now compute the wave speed: \(v = \sqrt{\frac{70.0 N}{0.015 kg/m}} = 68.41 m/s\)
03

Calculate the angular frequency and wave number

The angular frequency (ω) can be determined using the given frequency (f): \(ω = 2πf\) Given, frequency, \(f = 50.0 Hz\) Now, calculate the angular frequency: \(ω = 2π(50.0 Hz) = 100π~rad/s\) Next, calculate the wave number (k) using the wave speed (v) and angular frequency (ω): \(k = \frac{ω}{v}\) Wave number, \(k = \frac{100π~rad/s}{68.41~m/s} = \frac{50π}{34.205} rad/m\)
04

Calculate the power

Now, we can find the power (P) needed to generate the wave using the given amplitude (A), wave speed (v), and wave number (k): \(P = \frac{1}{2}µvω^2A^2\) Given amplitude, \(A = 4.00 cm = 0.0400 m\) Now, substitute the values: \(P = \frac{1}{2}(0.015 kg/m)(68.41 m/s)(100π rad/s)^2(0.0400 m)^2\) Upon calculating, we get: \(P = 50.58 W\) So, the power required to generate the traveling wave with a frequency of \(50.0 Hz\) and an amplitude of \(4.00 cm\) on the given string is approximately \(50.58 W\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Fans at a local football stadium are so excited that their team is winning that they start "the wave" in celebration. Which of the following four statements is (are) true? I. This wave is a traveling wave. II. This wave is a transverse wave. III. This wave is a longitudinal wave. IV. This wave is a combination of a longitudinal wave and a transverse wave. a) I and II c) III only e) I and III b) II only d) I and IV

Two waves traveling in opposite directions along a string fixed at both ends create a standing wave described by \(y(x, t)=1.00 \cdot 10^{-2} \sin (25 x) \cos (1200 t) .\) The string has a linear mass density of \(0.01 \mathrm{~kg} / \mathrm{m},\) and the tension in the string is supplied by a mass hanging from one end. If the string vibrates in its third harmonic, calculate (a) the length of the string, (b) the velocity of the waves, and (c) the mass of the hanging mass.

a) Starting from the general wave equation (equation 15.9 ), prove through direct derivation that the Gaussian wave packet described by the equation \(y(x, t)=(5.00 m) e^{-0.1(x-5 t)^{2}}\) is indeed a traveling wave (that it satisfies the differential wave equation). b) If \(x\) is specified in meters and \(t\) in seconds, determine the speed of this wave. On a single graph, plot this wave as a function of \(x\) at \(t=0, t=1.00 \mathrm{~s}, t=2.00 \mathrm{~s},\) and \(t=3.00 \mathrm{~s}\) c) More generally, prove that any function \(f(x, t)\) that depends on \(x\) and \(t\) through a combined variable \(x \pm v t\) is a solution of the wave equation, irrespective of the specific form of the function \(f\)

The tension in a 2.7 -m-long, 1.0 -cm-diameter steel cable \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}\right)\) is \(840 \mathrm{~N}\). What is the fundamental frequency of vibration of the cable?

A \(50.0-\mathrm{cm}\) -long wire with a mass of \(10.0 \mathrm{~g}\) is under a tension of \(50.0 \mathrm{~N}\). Both ends of the wire are held rigidly while it is plucked. a) What is the speed of the waves on the wire? b) What is the fundamental frequency of the standing wave? c) What is the frequency of the third harmonic?
See all solutions

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Read Explanation

Measurements

Read Explanation

Fluids

Read Explanation

Particle Model of Matter

Read Explanation

Atoms and Radioactivity

Read Explanation

Rotational Dynamics

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.

Sign-up for free