A standing wave is set up in a string of variable length and tension by a vibrator of variable frequency. Both ends of the string are fixed. When the vibrator has a frequency fA, in a string of length LA and under tension TA, nA antinodes are set up in the string.
(a) Write an expression for the frequency fA of a standing wave in terms of the number nA, length LA, tension TA, and linear density μA.
fA=
(b) If the length of the string is doubled to LB = 2LA, what frequency fB (written as a multiple of fA) will result in the same number of antinodes? Assume the tension and linear density are unchanged. Hint: Make a ratio of expressions for fB and fA.
fB=
(c) If the frequency and length are held constant, what tension TB will produce nA + 1 antinodes? (Use any variable or symbol stated above as necessary.)
TB=
(d) If the frequency is tripled and the length of the string is halved, by what factor should the tension be changed so that twice as many antinodes are produced?
(a) To find the expression for the frequency fA of a standing wave in terms of the number nA, length LA, tension TA, and linear density μA, 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, and μ is the linear density.
The speed of the wave can also be expressed as the product of the frequency and the wavelength:
v = fA * λ,
where fA is the frequency and λ is the wavelength.
In a standing wave, there is a fixed relationship between the length of the string and the number of antinodes or nodes. For a standing wave with nA antinodes, the length of the string is related to the wavelength by the formula:
LA = (nA * λ) / 2.
Rearranging this equation, we can express the wavelength λ in terms of the length of the string LA and the number of antinodes nA:
λ = 2 * LA / nA.
Substituting this expression for the wavelength into the equation for the speed of the wave, we get:
v = fA * (2 * LA / nA).
Equating this expression for the speed of the wave with the earlier expression (√(T/μ)):
fA * (2 * LA / nA) = √(T/μ).
Simplifying this equation, we can find the expression for fA:
fA = √((T * nA) / (2 * μ * LA)).
So the expression for the frequency fA of a standing wave in terms of the number nA, length LA, tension TA, and linear density μA is:
fA = √((TA * nA) / (2 * μA * LA)).
(b) To find the frequency fB (written as a multiple of fA) that will result in the same number of antinodes when the length of the string is doubled to LB = 2LA, we can use the relationship between the length and the number of antinodes in a standing wave.
For the same number of antinodes, the wavelength should remain the same when the length is doubled. Since the length is doubled, the wavelength must also double:
λB = 2 * λA.
Using the formula for the wavelength in terms of the length of the string:
λ = 2 * LA / nA,
we can express this relationship as:
(2 * LA) / nB = 2 * (2 * LA) / nA.
Simplifying this equation, we can find the relationship between the number of antinodes and the frequencies:
nB = 2 * nA.
Since the number of antinodes doubles, the frequency of the standing wave fB must be half of the original frequency fA:
fB = fA / 2.
So the frequency fB that will result in the same number of antinodes when the length of the string is doubled is equal to half of the frequency fA.
(c) If the frequency and length are held constant, to find the tension TB that will produce nA + 1 antinodes, we can start by rearranging the formula for the wavelength:
λ = 2 * LA / nA.
Since the length and frequency are held constant, the wavelength must also be constant:
λA = λB.
Using the formulas for the wavelength:
(2 * LA) / nA = (2 * LB) / (nA + 1).
Simplifying this equation, we can find the relationship between the number of antinodes and the lengths of the strings:
(2 * LA * (nA + 1)) = 2 * LA * nA.
Simplifying further, we find:
nA + 1 = nA.
This equation is not possible since it states that nA + 1 is equal to nA, which is incorrect. Therefore, there is no tension that can produce nA + 1 antinodes with a constant frequency and length.
(d) If the frequency is tripled and the length of the string is halved, we need to find the factor by which the tension should be changed to produce twice as many antinodes.
Let's define the original tension as TA and the original number of antinodes as nA.
Since the frequency is tripled and the length is halved, the wavelength is:
λB = λA / 2.
Using the formulas for the wavelength:
(2 * LA) / nA = 2 * LB / (2 * nA).
Simplifying this equation, we can find the relationship between the number of antinodes and the lengths of the strings:
LA = LB.
Since the lengths are equal, the wavelengths are also equal:
λA = λB.
Using the formulas for the wavelength:
(2 * LA) / nA = (2 * LB) / (2 * nA).
Simplifying this equation, we can find the relationship between the number of antinodes and the lengths of the strings:
nA = nA.
This equation is always true since it states that nA is equal to nA.
Therefore, if the frequency is tripled and the length of the string is halved, the tension should be changed by a factor of 4 (quadrupled) to produce twice as many antinodes.