A stretched string is 1.93 m long and has a mass of 21.1 g. When the string is oscillated at 440 Hz, which is the frequency of the standard A pitch that orchestras tune to, transverse waves with a wavelength of 16.9 cm
travel along the string. Calculate the tension in the string.
I have been using the formula v = sqrt (T/(m/L)
which would give T = (440x0.169)^2 (0.211/1.93)
why isn't this the correct way to solve?
The formula you used, v = sqrt(T/(m/L)), is for calculating the velocity of a wave on a string. However, in this case, you are asked to calculate the tension in the string.
To calculate the tension in the string, you can use the formula:
v = f * λ
where:
v is the velocity of the wave,
f is the frequency of the wave, and
λ is the wavelength of the wave.
First, let's convert the wavelength to meters:
λ = 16.9 cm = 0.169 m
Now, rearrange the formula to solve for the velocity:
v = f * λ
v = 440 Hz * 0.169 m
v ≈ 74.36 m/s
Since the velocity of the wave is given by the formula:
v = sqrt(T/(m/L))
We can rearrange this formula to solve for T (the tension in the string):
T = (m/L) * v^2
Let's substitute the values into the formula:
T = (0.211 g / 1.93 m) * (74.36 m/s)^2
Now, we need to convert the mass from grams to kilograms:
0.211 g = 0.211 × 10^-3 kg
Substitute the values and calculate:
T = (0.211 × 10^-3) kg / 1.93 m * (74.36 m/s)^2
T ≈ 0.157 N
Therefore, the tension in the string is approximately 0.157 N.
The formula you're using, v = sqrt(T/(m/L)), is the correct formula for the velocity of waves on a string. However, to determine the tension in the string, we need to use a different formula that incorporates the wave velocity and wavelength.
The correct formula to calculate the tension in a string with waves is:
T = (μ * v^2) / L
Where:
T is the tension in the string
μ is the linear mass density of the string (mass per unit length)
v is the velocity of the wave on the string
L is the length of the string
In this case, we are given the length of the string (L = 1.93 m), the mass of the string (m = 21.1 g), and the wavelength of the wave on the string (λ = 16.9 cm = 0.169 m). From the given frequency (f = 440 Hz), we can calculate the velocity of the wave using the formula v = f * λ.
First, convert the mass of the string from grams to kilograms:
m = 21.1 g = 0.0211 kg
Next, calculate the velocity of the wave:
v = f * λ
= 440 Hz * 0.169 m
= 74.36 m/s
Now, to calculate the linear mass density (μ), we divide the mass by the length of the string:
μ = m / L
= 0.0211 kg / 1.93 m
≈ 0.0109 kg/m
Finally, we can now plug the values into the tension formula:
T = (μ * v^2) / L
= (0.0109 kg/m * (74.36 m/s)^2) / 1.93 m
≈ 27.03 N
Therefore, the tension in the string is approximately 27.03 Newtons.