2. A rope has a length of 86cm. If the fundamental frequency is 230 Hz, determine and draw the first 3 harmonics.

Here's how to determine and draw the first three harmonics of a vibrating rope:<br /><br />**Understanding Harmonics**<br /><br />Harmonics are the natural frequencies at which a rope (or other vibrating object) can oscillate. The fundamental frequency (also called the first harmonic) is the lowest frequency. Higher harmonics are integer multiples of the fundamental frequency.<br /><br />* **1st Harmonic (Fundamental):** The rope vibrates in one segment.<br />* **2nd Harmonic:** The rope vibrates in two segments.<br />* **3rd Harmonic:** The rope vibrates in three segments.<br /><br />**Calculations**<br /><br />* **Fundamental Frequency (f₁):** Given as 230 Hz.<br />* **2nd Harmonic (f₂):** f₂ = 2 * f₁ = 2 * 230 Hz = 460 Hz<br />* **3rd Harmonic (f₃):** f₃ = 3 * f₁ = 3 * 230 Hz = 690 Hz<br /><br />**Wavelengths**<br /><br />The relationship between the wavelength (λ), frequency (f), and the speed of the wave (v) is: v = fλ. We don't know the speed of the wave on the rope, but we *do* know that for each harmonic, a certain number of half-wavelengths fit along the length of the rope (L).<br /><br />* **1st Harmonic:** L = λ₁/2 => λ₁ = 2L = 2 * 86 cm = 172 cm<br />* **2nd Harmonic:** L = λ₂ => λ₂ = L = 86 cm<br />* **3rd Harmonic:** L = (3/2)λ₃ => λ₃ = (2/3)L = (2/3) * 86 cm ≈ 57.3 cm<br /><br />**Drawings**<br /><br />Here's how to draw the first three harmonics:<br /><br />1. **1st Harmonic:** Draw a smooth curve resembling a single, large "U" shape along the length of the rope. The ends of the rope are fixed points (nodes), and the center has maximum displacement (antinode).<br /><br /> ```<br /> _<br /> / \<br /> | |<br /> \_/<br /> ```<br /><br />2. **2nd Harmonic:** Draw two smaller "U" shapes connected end-to-end. There will be nodes at the ends and in the middle of the rope, and antinodes between the nodes.<br /><br /> ```<br /> _ _<br /> / \ / \<br /> | | |<br /> \_/ \_/<br /> ```<br /><br />3. **3rd Harmonic:** Draw three even smaller "U" shapes connected end-to-end. There will be nodes at the ends and at two points along the rope, dividing it into thirds.<br /><br /> ```<br /> _ _ _<br /> / \ / \ / \<br /> | | | |<br /> \_/ \_/ \_/<br /> ```<br /><br />**Key Points in the Drawings**<br /><br />* **Nodes:** Points of zero displacement (the ends of the rope and the points where the wave crosses the horizontal axis).<br />* **Antinodes:** Points of maximum displacement (the peaks and troughs of the waves).<br />* The wavelength is the distance between two consecutive nodes or two consecutive antinodes.<br /><br /><br />This provides a complete explanation of how to calculate and draw the first three harmonics of a vibrating rope. Remember to label your drawings with the harmonic number and frequency.<br />