A tight uniform string with a length of 1.80 m is tied down at both ends and placed under a tension of 100 N. When it vibrates in its third harmonic, the sound given off has a frequency of 79.0Hz .
Part A
What is the mass of the string?
Velocity in the stretched string is
v = sqrt(TL/m)
For the 3rd harmonic λ=2L/3
λ=v/f => v= λf=2Lf/3
sqrt(TL/m)= 2Lf/3
TL/m=(2Lf/3)²
m=9TL/4L²f²=9T/4Lf²=
=9•100/4•1.8•79²=0.02 kg=20 g
To find the mass of the string, we can use the formula:
m = (ρ * A * L) / (n * T²)
where:
m = mass of the string
ρ = density of the string
A = cross-sectional area of the string
L = length of the string
n = harmonic number (3 in this case)
T = tension in the string
Let's calculate it step by step:
Step 1: Identify the given values:
L = 1.80 m
n = 3
T = 100 N
Step 2: Determine the properties of the string:
To calculate the mass, we need the density (ρ) and the cross-sectional area (A) of the string. These properties will depend on the material of the string.
Step 3: Substitute the given values into the formula:
m = (ρ * A * L) / (n * T²)
Step 4: Solve for the mass (m):
Since we don't have the density (ρ) and cross-sectional area (A) of the string, we cannot calculate the mass without this information.
To find the mass, we need either the density and cross-sectional area of the string or additional information about the string material.
To find the mass of the string, we need to use the formula relating the speed of a wave on a string to its tension and linear mass density.
The formula is:
v = sqrt(T/μ)
Where:
v is the speed of the wave
T is the tension in the string
μ is the linear mass density (mass per unit length) of the string
First, let's find the speed of the wave using the frequency and the wavelength. In the third harmonic, we know that the wavelength is equal to twice the length of the string.
λ = 2L = 2 * 1.80 m = 3.60 m
The speed of the wave can be calculated by:
v = f * λ
v = 79.0 Hz * 3.60 m = 284.4 m/s
Now, we can rearrange the formula to solve for μ.
μ = T / v^2
Plug in the values:
μ = 100 N / (284.4 m/s)^2
Calculating this, we find:
μ ≈ 0.00122 kg/m
The linear mass density (mass per unit length) of the string is approximately 0.00122 kg/m.
Now, to find the mass of the string, we can multiply the linear mass density by the length of the string:
mass = μ * length
mass = 0.00122 kg/m * 1.80 m
Calculating this, we find:
mass ≈ 0.0022 kg
Therefore, the mass of the string is approximately 0.0022 kg.