A 3.0-g string, 0.14 m long, is under tension. The string produces a 200-Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. Find the tension in the string.
4.8N
Why did the string go to therapy? Because it had tension issues! Let's find the tension in the string.
To solve this problem, we can use the equation for the speed of a wave on a string:
v = √(F/μ)
where v is the speed of the wave, F is the tension in the string, and μ is the linear mass density of the string.
The linear mass density of the string can be calculated using the equation:
μ = m/L
where m is the mass of the string and L is the length of the string.
Given:
m = 3.0 g = 0.003 kg (convert grams to kilograms)
L = 0.14 m
v = 344 m/s (speed of sound in air)
f = 200 Hz (frequency of the tone)
To find the tension (F), we need to first find μ using the given data:
μ = m/L = 0.003 kg / 0.14 m = 0.02143 kg/m
Now we can use the given frequency and the speed of sound to find the wavelength (λ) of the wave:
λ = v/f = 344 m/s / 200 Hz = 1.72 m
Since the string is vibrating in the third harmonic, the wavelength is given by:
λ = 2L/3
Simplifying the equation, we find:
2L/3 = 1.72 m
Now we can solve for L:
2L = 3(1.72 m)
2L = 5.16 m
L = 2.58 m
Now that we have L, we can substitute the values into the equation for μ:
μ = m/L = 0.003 kg / 2.58 m = 0.00116 kg/m
Now let's substitute the values into the equation for the wave speed:
v = √(F/μ)
Using the given value for v, we can solve for F:
344 m/s = √(F/0.00116 kg/m)
Squaring both sides of the equation, we find:
118336 = F / 0.00116
Solving for F, we get:
F = 118336 × 0.00116 = 137.2416 N
Therefore, the tension in the string is approximately 137.24 N. Now that's one melodious string!
To find the tension in the string, we can use the formula for the speed of waves on a string:
v = √(T/μ)
where v is the speed of the wave, T is the tension in the string, and μ is the linear density of the string.
Step 1: Calculate the linear density of the string:
The linear density (μ) is given by the mass per unit length. We are given the mass of the string (3.0 g) and its length (0.14 m). We divide the mass by the length to find the linear density.
μ = m/ℓ
where:
m = mass of the string = 3.0 g = 3.0 × 10⁻³ kg (converting from grams to kilograms)
ℓ = length of the string = 0.14 m
μ = 3.0 × 10⁻³ kg / 0.14 m
Step 2: Calculate the speed of the wave:
The speed of the wave (v) is given as 344 m/s.
Step 3: Calculate the frequency of the wave:
The frequency of the wave (f) is given as 200 Hz.
Step 4: Determine the relationship between frequency, speed, and wavelength:
We know that the frequency, speed, and wavelength (λ) of a wave are related by the equation:
v = fλ
where:
v = speed of the wave (344 m/s)
f = frequency of the wave (200 Hz)
λ = wavelength of the wave
Step 5: Determine the wavelength of the wave:
Since the string is vibrating in the third harmonic, we know that the wavelength is three times the length of the string.
λ = 3ℓ
where:
ℓ = length of the string = 0.14 m
Step 6: Substitute the values into the equation to find the wavelength:
From step 5, we have:
λ = 3 × 0.14 m
Step 7: Calculate the tension in the string:
Now we can substitute the values of v and μ into the formula from step 7 and solve for T.
v = √(T/μ)
T = μv²
T = (3.0 × 10⁻³ kg / 0.14 m) × (344 m/s)²
Simplify the expression and calculate to find the tension in the string.
To find the tension in the string, you can use the equation for the wave speed in a string:
v = sqrt(T/μ)
Where:
v is the wave speed in the string,
T is the tension in the string, and
μ is the linear mass density of the string.
To find the tension, we need to first determine the linear mass density of the string, and then substitute it into the equation along with the wave speed.
The linear mass density (μ) of a string is given by:
μ = m/L
Where:
m is the mass of the string, and
L is the length of the string.
In this case, the mass of the string is given as 3.0 g, or 0.003 kg, and the length of the string is given as 0.14 m.
μ = 0.003 kg / 0.14 m
μ ≈ 0.0214 kg/m
Now, we can substitute the given wave speed (344 m/s) and the calculated linear mass density (0.0214 kg/m) into the wave speed equation:
344 m/s = sqrt(T / 0.0214 kg/m)
Squaring both sides of the equation:
118336 = T / 0.0214
Now solve for T:
T = 118336 * 0.0214
T ≈ 2532.4 N
Therefore, the tension in the string is approximately 2532.4 Newtons.