Answer :
The wavelengths and frequencies of the first four modes of standing waves on the string are approximately: Mode 1 - λ = 6.70 m, f = 120.6 Hz; Mode 2 - λ = 3.35 m, f = 241.2 Hz; Mode 3 - λ ≈ 2.23 m, f ≈ 362.2 Hz; Mode 4 - λ = 3.35 m, f = 241.2 Hz.
To find the wavelengths and frequencies of the first four modes of standing waves on the string, we can use the formula:
λ = 2L/n
Where:
λ is the wavelength,
L is the length of the string, and
n is the mode number.
The frequencies can be calculated using the formula:
f = v/λ
Where:
f is the frequency,
v is the wave speed (determined by the tension and mass per unit length of the string), and
λ is the wavelength.
Given:
Mass of the string (m) = 0.0010 kg
Length of the string (L) = 3.35 m
Tension (T) = 195 N
First, we need to calculate the wave speed (v) using the formula:
v = √(T/μ)
Where:
μ is the linear mass density of the string, given by μ = m/L.
μ = m/L = 0.0010 kg / 3.35 m = 0.0002985 kg/m
v = √(195 N / 0.0002985 kg/m) = √(652508.361 N/m^2) ≈ 808.03 m/s
Now, we can calculate the wavelengths (λ) and frequencies (f) for the first four modes (n = 1, 2, 3, 4):
For n = 1:
λ₁ = 2L/1 = 2 * 3.35 m = 6.70 m
f₁ = v/λ₁ = 808.03 m/s / 6.70 m ≈ 120.6 Hz
For n = 2:
λ₂ = 2L/2 = 3.35 m
f₂ = v/λ₂ = 808.03 m/s / 3.35 m ≈ 241.2 Hz
For n = 3:
λ₃ = 2L/3 ≈ 2.23 m
f₃ = v/λ₃ = 808.03 m/s / 2.23 m ≈ 362.2 Hz
For n = 4:
λ₄ = 2L/4 = 3.35 m
f₄ = v/λ₄ = 808.03 m/s / 3.35 m ≈ 241.2 Hz
Therefore, the wavelengths and frequencies of the first four modes of standing waves on the string are approximately:
Mode 1: Wavelength (λ) = 6.70 m, Frequency (f) = 120.6 Hz
Mode 2: Wavelength (λ) = 3.35 m, Frequency (f) = 241.2 Hz
Mode 3: Wavelength (λ) ≈ 2.23 m, Frequency (f) ≈ 362.2 Hz
Mode 4: Wavelength (λ) = 3.35 m, Frequency (f) = 241.2 Hz
To know more about frequency refer here
https://brainly.com/question/29739263#
#SPJ11
