The transverse standing wave on a string fixed at both ends is vibrating at its fundamental frequency of 250 Hz. What would be the fundamental frequency on a piece of the same string that is twice as long and has four times the tension?
I know the equation f=v/2L. and that v=(F/(m/L))^(1/2). But i don't know where to go from there.
tension 4x, so v must be 2x. Now, string twice as long halves freq, right. So the freq is 2x/2x=same freq.
check my reasoning.
Wow, seems like you've got all the fancy equations! Don't worry, I'm here to help you tackle this puzzler.
To find the fundamental frequency of the new string, let's break it down step by step.
First, we know that the fundamental frequency, f₁, of the original string is 250 Hz.
Next, let's consider the new string. It is twice as long as the original string, so its length, L₂, would be 2 times L.
Also, the new string has four times the tension, so the tension, F₂, would be 4 times F.
Now, let's plug these values into the equation f₂ = v₂ / 2L₂.
But wait! We need to figure out the velocity, v₂, of the new string as well. Don't worry, I got your back.
Using the equation v=(F/(m/L))^(1/2), we can see that the velocity is directly proportional to the square root of tension and inversely proportional to the square root of mass per unit length.
Since we're assuming the mass per unit length of the string remains the same, the velocity is only affected by the tension.
Given that the tension in the new string is four times greater, the velocity, v₂, would be twice that of the original string. So, v₂ = 2v.
Now we can go back to our previous equation f₂ = v₂ / 2L₂ and plug in the values:
f₂ = 2v / (2 × 2L).
Phew! Simplifying that, we get:
f₂ = v / 4L.
But remember that f₁ = 250 Hz. So, we can substitute that in:
250 Hz = v / 4L.
Rearranging the equation to solve for v, we get:
v = 1000L Hz.
And finally, plugging that value back into f₂ = v / 4L, we get:
f₂ = 1000L / 4L.
Which simplifies to:
f₂ = 250 Hz.
Yup, you got it right, even with all the clowning around! The fundamental frequency on the new string that's twice as long and has four times the tension would still be 250 Hz!
To find the fundamental frequency on a string that is twice as long and has four times the tension, we can use the formulas and equations you mentioned.
Given:
Fundamental frequency of the original string (f1) = 250 Hz
Length of the original string (L1) = L
Tension of the original string (T1) = T
We know that the velocity of the transverse wave on a string is given by:
v = √(F/(m/L))
The linear mass density (m/L) of the string remains constant. Therefore, we can say that m1/L1 = m2/L2, where m1 and m2 are the linear mass densities of the original and new strings, respectively.
Now, let's consider the new string:
Length of the new string (L2) = 2L (twice as long as the original string)
Tension of the new string (T2) = 4T (four times the tension of the original string)
To find the linear mass density (m2/L2), we can rearrange the given equation as:
m2/L2 = m1/L1
The velocity (v) on both strings will be the same, so:
v1 = v2
Plug in the formulas for velocity and linear mass density:
√(T1/(m1/L1)) = √(T2/(m2/L2))
Substituting T1, T2, L1, L2 based on the given values:
√(T/(m1/L)) = √((4T)/(m2/(2L)))
Now, simplify the equation:
√(T/(m1/L)) = √(4T/(m2/(2L)))
√(T/(m1/L)) = √(T/(m2/L))
√(T/(m1/L)) = √(T/(m2/L))
Since both sides of the equation are the same, we can equate them:
T/(m1/L) = T/(m2/L)
Now, solve for m2/L:
m2/L = T/(T/(m1/L))
m2/L = m1/L
The linear mass density of the new string (m2/L) is equal to the linear mass density of the original string (m1/L). Therefore, the fundamental frequency on the new string, f2, will be the same as the fundamental frequency of the original string, f1.
Hence, the fundamental frequency on the new string that is twice as long and has four times the tension will also be 250 Hz.
To find the fundamental frequency on a string that is twice as long and has four times the tension, you need to calculate the wave speed (v) of the new string and then use the equation f = v/2L to determine the frequency.
Let's break down the steps:
1. Calculate the wave speed (v) for the new string:
- The wave speed formula is v = √(F / μ), where F represents the tension and μ represents mass per unit length.
- Since the new string has four times the tension, the tension (F') on the new string is equal to 4F.
- The mass per unit length (μ') of the new string is equal to the mass per unit length (μ) of the original string.
- Substitute the values into the wave speed formula: v' = √(4F / μ).
2. Determine the fundamental frequency (f') on the new string:
- Use the equation f' = v' / (2L'), where L' is the length of the new string and v' is the wave speed of the new string.
- The length of the new string (L') is twice the length of the original string (L), so L' = 2L.
- Substitute the values into the equation: f' = (√(4F / μ)) / (2(2L)). Simplify the equation.
By following these steps, you can obtain the fundamental frequency for the new string based on its length and tension.