The frequency of a string with each property respectively is 800 Hz, 200 Hz and 400 Hz.
An equation is an expression that shows the relationship between two or more numbers and variables.
Let f represent the frequency, l represent the length, T represent the tension and m represent the mass density.
The fundamental frequency (in hertz) of a piano string is directly proportional to the square root of its tension and inversely proportional to its length and the square root of its mass density.
Hence:
f = k[√T / (l * √m)]
f = 400, hence:
400 = k[√T / (l * √m)]
k = 400[(l * √m) / √T]
a) For four times the tension:
f = 400[(l * √m) / √T] * [√(4T) / (l * √m)] = 400 * 2 = 800 Hz
b) Twice the length
f = 400[(l * √m) / √T] * [√(T) / (2l * √m)] = 400 / 2 = 200 Hz
a) For four times and twice the length:
f = 400[(l * √m) / √T] * [√(4T) / (2l * √m)] = 400 * 2 / 2 = 400 Hz
The frequency of a string with each property respectively is 800 Hz, 200 Hz and 400 Hz.
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