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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 \(50.58 W\).

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Most popular questions from this chapter

If two traveling waves have the same wavelength, frequency, and amplitude and are added appropriately, the result is a standing wave. Is it possible to combine two standing waves in some way to give a traveling wave?

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A wave traveling on a string has the equation of motion \(y(x, t)=0.02 \sin (5.00 x-8.00 t)\) a) Calculate the wavelength and the frequency of the wave. b) Calculate its velocity. c) If the linear mass density of the string is \(\mu=0.10 \mathrm{~kg} / \mathrm{m}\), what is the tension on the string?

A traveling wave propagating on a string is described by the following equation: $$ y(x, t)=(5.00 \mathrm{~mm}) \sin \left(\left(157.08 \mathrm{~m}^{-1}\right) x-\left(314.16 \mathrm{~s}^{-1}\right) t+0.7854\right) $$ a) Determine the minimum separation, \(\Delta x_{\min }\), between two points on the string that oscillate in perfect opposition of phases (move in opposite directions at all times). b) Determine the separation, \(\Delta x_{A B}\), between two points \(A\) and \(B\) on the string, if point \(B\) oscillates with a phase difference of 0.7854 rad compared to point \(A\). c) Find the number of crests of the wave that pass through point \(A\) in a time interval \(\Delta t=10.0 \mathrm{~s}\) and the number of troughs that pass through point \(B\) in the same interval. d) At what point along its trajectory should a linear driver connected to one end of the string at \(x=0\) start its oscillation to generate this sinusoidal traveling wave on the string?
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