The A string on a string bass is turned to vibrate at a fundamental frequency of 55.0Hz. If the tension in the string were increased by a factor of four, what would be the new fundamental frequency?
I need some guides to do it. Thx.
frequency proportional to square root of tension
sqrt 4 = 2
so 2*55 = 110
To find the new fundamental frequency, we need to understand the relationship between tension and frequency in a vibrating string.
1. First, let's consider the relationship between the tension (T), the length of the string (L), and the mass per unit length of the string (µ). The fundamental frequency (f) can be determined using the equation:
f = (1/2L) * sqrt(T/µ)
2. The equation tells us that frequency is directly proportional to the square root of tension. So, if we increase the tension by a factor of four, the new tension (T') will be four times the original tension (T): T' = 4T.
3. Applying the new tension (T') in the frequency equation, we get:
f' = (1/2L) * sqrt(T'/µ)
= (1/2L) * sqrt(4T/µ)
= (1/2L) * 2 * sqrt(T/µ)
= (1/2L) * 2 * sqrt(T) * sqrt(1/µ)
= (1/2L) * 2 * sqrt(T) * (1/sqrt(µ))
= (1/2L) * 2 * sqrt(T) * (1/õ)
4. We can simplify the equation further. Square roots of ratios can be represented as the ratios of square roots:
f' = (1/2L) * 2 * sqrt(T) * (1/sqrt(µ))
= (1/2L) * 2 * (sqrt(T)/sqrt(µ))
= (1/2L) * 2 * √(T/µ)
= (1/2L) * 2 * √T' [since T' = 4T]
5. Therefore, the new fundamental frequency (f') can be expressed as:
f' = (1/2L) * 2 * √T'
In summary, to find the new fundamental frequency when the tension is increased by a factor of four, we need to multiply the original fundamental frequency by the square root of four (which is two).
the laws of strings applies:
http://primes.utm.edu/mersenne/LukeMirror/mirrors/eb/mers_02.htm
What is the square root of four?