Problem 34 - University Physics with Modern Physics

Question

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 \(4.00 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

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 1: 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: \(µ = \frac{0.030 kg}{2.00 m} = 0.015 kg/m\)

Step 2: 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: \(v = \sqrt{\frac{70.0 N}{0.015 kg/m}} = 68.41 m/s\)

Step 3: 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, \(k = \frac{100π~rad/s}{68.41~m/s} = \frac{50π}{34.205} rad/m\)

Step 4: 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 \(50.58 W\).