Let's break down the problem step by step. ### **Given Data** - **Mass of string (m):** .001 kg - **Length of string (L):** 4.6 m - **Tension (T):** 180 N We are to find the **wavelengths** and **frequencies** of the first four modes (n = 1, 2, 3, 4). --- ## **Step 1: Calculate the Linear Mass Density** \[ \mu = \frac{m}{L} = \frac{.001}{4.6} \approx 2.174 \times 10^{-4}\ \text{kg/m} \] --- ## **Step 2: Find the Wave Speed** \[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{180}{2.174 \times 10^{-4}}} \] \[ v \approx \sqrt{827,921.7} \approx 910.97\ \text{m/s} \] --- ## **Step 3: Wavelengths and Frequencies for Modes** The wavelength for the n-th harmonic (mode): \[ \lambda_n = \frac{2L}{n} \] The frequency for the n-th harmonic: \[ f_n = \frac{v}{\lambda_n} \] --- ## **Step 4: Calculate for n = 1, 2, 3, 4** ### **1st Mode (n=1):** \[ \lambda_1 = \frac{2 \times 4.6}{1} = 9.2\ \text{m} \] \[ f_1 = \frac{910.97}{9.2} \approx 99.02\ \text{Hz} \] ### **2nd Mode (n=2):** \[ \lambda_2 = \frac{2 \times 4.6}{2} = 4.6\ \text{m} \] \[ f_2 = \frac{910.97}{4.6} \approx 198.03\ \text{Hz} \] ### **3rd Mode (n=3):** \[ \lambda_3 = \frac{2 \times 4.6}{3} \approx 3.07\ \text{m} \] \[ f_3 = \frac{910.97}{3.07} \approx 296.74\ \text{Hz} \] ### **4th Mode (n=4):** \[ \lambda_4 = \frac{2 \times 4.6}{4} = 2.3\ \text{m} \] \[ f_4 = \frac{910.97}{2.3} \approx 396.07\ \text{Hz} \] --- ## **Final Answers** | Mode | Wavelength (\(\lambda\)) | Frequency (\(f\)) | |------|--------------------------|-------------------| | n=1 | 9.2 m | 99.02 Hz | | n=2 | 4.6 m | 198.03 Hz | | n=3 | 3.07 m | 296.74 Hz | | n=4 | 2.3 m | 396.07 Hz | --- ### **Boxed Answers** - **a.** \(\lambda_1 = 9.2\) m, \(f_1 = 99.02\) Hz - **b.** \(\lambda_2 = 4.6\) m, \(f_2 = 198.03\) Hz - **c.** \(\lambda_3 = 3.07\) m, \(f_3 = 296.74\) Hz - **d.** \(\lambda_4 = 2.3\) m, \(f_4 = 396.07\) Hz