A standing wave is formed on a string for which three and one half wavelengths fit into the total length of the string, which is 2.5 m. The wave has a period of 0.10 s. If the string has a mass density of 220 g/m, what is the tension in the string? (Answer: 11.2N)
Answer is provided above. Please show all work on how to get the answer. Thank you!
calculatae wavelength: 2.5m/3.5
Law of tensions:
velocity=sqrt(tension/mu)
find velocity: v=f*wavelength=wavelength/period
mu given at .22kg/m
tension=v^2*mu=work it out
To find the tension in the string, we can use the formula for the speed of a wave on a string:
v = √(T/μ)
where v is the speed of the wave, T is the tension in the string, and μ (mu) is the mass density of the string.
We can also find the speed of the wave using the formula:
v = λ/T
where λ (lambda) is the wavelength of the wave, and T is the period of the wave.
Given that three and a half wavelengths fit into the total length of the string (2.5 m), we can find the wavelength:
λ = L/n
where L is the total length of the string and n is the number of wavelengths.
Using this information, we can find the wavelength of the wave:
λ = 2.5 m / 3.5 = 0.714 m
Now that we have the wavelength and period of the wave, we can find the speed:
v = λ/T = 0.714 m / 0.10 s = 7.14 m/s
We also know the mass density of the string (220 g/m). To use this value in the equation for the speed of the wave, we need to convert it to kg/m:
μ = 220 g/m * (1 kg / 1000 g) = 0.22 kg/m
Now we have all the information we need to solve for the tension in the string. Rearranging the equation for the speed of the wave:
v = √(T/μ)
T/μ = v^2
T = μ * v^2
Substituting the values:
T = 0.22 kg/m * (7.14 m/s)^2 = 11.2 N
Therefore, the tension in the string is 11.2 N.
To find the tension in the string, we can use the wave equation:
v = λ * f
where:
v is the velocity of the wave,
λ is the wavelength,
and f is the frequency of the wave.
First, let's solve for the wavelength (λ):
In this problem, we are given that three and one half wavelengths fit into the total length of the string, which is 2.5 m.
So, the length of one wavelength (λ) is:
λ = 2.5 m / (3.5 wavelengths) = 0.714 m
Next, let's find the velocity (v) by rearranging the wave equation:
v = λ * f
We are given the period (T), and since the period (T) is the reciprocal of the frequency (f), we can use the equation:
f = 1 / T
Given that the period (T) is 0.10 s, we can substitute it into the equation to find the frequency (f):
f = 1 / 0.10 s = 10 Hz
Now, we can find the velocity (v):
v = λ * f = 0.714 m * 10 Hz = 7.14 m/s
Finally, let's find the tension (T) in the string by using the equation:
T = μ * v^2
where:
μ is the mass density of the string,
v is the velocity of the wave.
Given that the mass density (μ) is 220 g/m, we need to convert it to kg/m:
μ = 220 g/m * (1 kg / 1000 g) = 0.220 kg/m
Now, we can plug in the values to find the tension (T):
T = (0.220 kg/m) * (7.14 m/s)^2 = 11.2 N
Therefore, the tension in the string is 11.2N.