A 1.6-m-long string fixed at both ends vibrates at resonant frequencies of 792 Hz and 990 Hz, with no other resonant frequency between these values.
(a) What is the fundamental frequency of this string?
(b) When the tension in the string is 1165 N, what is the total mass of the string?
Honestly, I am confused about this. COuld someone explain it to me. Thank you.
Well, let me "string" together an explanation for you!
(a) The fundamental frequency of a vibrating string is the lowest resonant frequency that it can produce. Since there are no other resonant frequencies between 792 Hz and 990 Hz, the fundamental frequency must be closer to 792 Hz. So, we can conclude that the fundamental frequency of this string is 792 Hz.
(b) To find the total mass of the string, we need to consider the tension in the string. The relationship between tension (T), mass per unit length (μ), and wave speed (v) in a string is given by the formula v = √(T/μ).
Since we know the tension (1165 N) and we need to find the mass per unit length (μ), we can rearrange the formula to solve for μ. Squaring both sides of the equation gives us v^2 = T/μ, and rearranging further gives us μ = T / v^2.
To find the wave speed (v), we need to use the formula v = fλ, where f is the frequency and λ is the wavelength. Since we know the frequency (792 Hz), and the wavelength for the fundamental frequency is twice the length of the string (1.6 m), we can calculate the wave speed.
v = fλ = 792 Hz x 1.6 m x 2 = 2534.4 m/s
Now we can substitute the values into the formula to find the mass per unit length (μ):
μ = T / v^2 = 1165 N / (2534.4 m/s)^2
Once you calculate this value, you'll have the mass per unit length of the string. To find the total mass, you can multiply the mass per unit length by the length of the string (1.6 m).
Voila! That should help you get a "grip" on the problem. If you have any more questions or need a little more "string" assistance, feel free to ask!
Sure! I can help you understand this problem step-by-step.
(a) To find the fundamental frequency of the string, we need to know the length of the string. In this case, the length of the string is given as 1.6 meters.
The fundamental frequency of a vibrating string is determined by its length and the speed of the wave traveling through it. The formula for calculating the fundamental frequency is:
f = v / (2L)
Where:
f = fundamental frequency
v = speed of the wave
L = length of the string
In this case, we know the length of the string (L = 1.6m), but we don't have the speed of the wave. However, we can determine the wave speed by using the relationship between frequency, wavelength, and wave speed:
v = f × λ
Where:
v = wave speed
f = frequency
λ = wavelength
Since the string is fixed at both ends, the wavelength of the fundamental frequency is twice the length of the string:
λ = 2L
Plugging this value for λ back into the equation for wave speed, we have:
v = f × 2L
Now we can substitute this expression for the wave speed back into the equation for the fundamental frequency:
f = (f × 2L) / (2L)
Simplifying the equation, we find:
f = f / 2
So the fundamental frequency (f1) is equal to half of the lowest resonant frequency:
f1 = 792 Hz / 2 = 396 Hz
Therefore, the fundamental frequency of the string is 396 Hz.
(b) To find the total mass of the string when the tension is 1165 N, we need to use the equation for wave speed:
v = √(T / μ)
Where:
v = wave speed
T = tension in the string
μ = linear mass density
In this case, we are given the tension in the string (T = 1165 N), but we need to find the linear mass density (μ).
The linear mass density can be calculated using the formula:
μ = m / L
Where:
μ = linear mass density
m = total mass of the string
L = length of the string
Rearranging this equation, we can solve for the total mass of the string:
m = μ × L
Now, we can substitute the expression for the linear mass density back into the equation for wave speed:
v = √(T / (m / L))
Simplifying further:
v = √(T × L / m)
Now, we can rearrange this equation to solve for the total mass of the string:
m = T × L / v²
Plugging in the given values, we can find the total mass of the string:
m = 1165 N × 1.6 m / (v²)
However, we don't have the value for the wave speed (v). Therefore, we can't calculate the total mass of the string with the given information.
Sure! I'd be happy to explain it to you.
To find the fundamental frequency of the string, we need to understand the concept of harmonics. In vibrating systems like a string, resonance occurs at frequencies where the length of the medium is an integer multiple of half the wavelength of the wave produced. The fundamental frequency corresponds to the first harmonic, which is the lowest possible frequency at which resonance occurs.
(a) To calculate the fundamental frequency, we can use the formula:
Fundamental frequency = (Speed of wave) / (2 × Length of string)
We know the length of the string is 1.6 m, but we need to determine the speed of the wave. We can do this by finding the average of the known resonant frequencies:
Average frequency = (792 Hz + 990 Hz) / 2
Now, we can calculate the fundamental frequency:
Fundamental frequency = Average frequency / 2 = (792 Hz + 990 Hz) / 2 / 2
(b) To find the total mass of the string when the tension is 1165 N, we need to use the formula for wave speed:
Wave speed = √(tension / linear mass density)
Given the tension in the string is 1165 N, we need to determine the linear mass density. Unfortunately, we don't have enough information to directly calculate it.
However, we know that the resonant frequencies are inversely proportional to the length of the string. Hence, if we increase the length of the string, the fundamental frequency will decrease. Conversely, if we decrease the length, the frequency will increase.
Since there is no other resonant frequency between 792 Hz and 990 Hz, we can conclude that the 792 Hz frequency corresponds to the first harmonic (fundamental frequency) and the 990 Hz frequency corresponds to the second harmonic.
Knowing this, we can set up the following relationship:
Fundamental frequency = (Speed of wave) / (2 × Length of string)
Second harmonic frequency = 2 × Fundamental frequency
Substituting the known values:
792 Hz = (Speed of wave) / (2 × 1.6 m)
990 Hz = (Speed of wave) / (2 × 1.6 m)
From these equations, we can solve for the speed of the wave. Then, using the obtained wave speed and the given tension of 1165 N, we can calculate the linear mass density.
Once we have the linear mass density, we can determine the total mass of the string by multiplying it by the length of the string.
I hope this explanation helps you understand the problem better!
f1 is the fundamental frequency. Resonant frequencies are
f(n) = n•f1 = 792 Hz,
f(n+1) =(n+1) •f1= 990 Hz
f(n)/f(n+1) = n/(n+1) = 792/990.
n = 4,
f1 =f(n)/n = 792/4 = 198 Hz,
f1 = v/2•L ,
v = f1•2•L = 198•2•1.6 = 633.6 m/s,
v = sqrt(T/m(o)),
m(o) = m/L,
m =T•L/v^2 = 1165•1.6/(633.6)^2 = 4.23•10^-3 kg =4.23 g.