A guitar string of length 30 cm and stretched under a tension of 78 N has a certain fundamental frequency. How long would a pipe, open at both ends, need to be to play the same fundamental frequency? A 15-cm long piece of the guitar string has a mass of 0.4 g. The speed of sound in air is 340 m/s.
To find the length of a pipe open at both ends that would produce the same fundamental frequency as a guitar string, we can use the equation for the fundamental frequency of an open pipe:
f = v / (2L)
Where:
f is the fundamental frequency of the pipe,
v is the speed of sound in air, and
L is the length of the pipe.
Given the length of the guitar string (30 cm) and the speed of sound in air (340 m/s), we need to convert the length of the guitar string from centimeters to meters:
L_guitar = 30 cm = 0.3 m
Now, let's find the fundamental frequency of the guitar string using the tension in the string and its mass:
m = 0.4 g = 0.4 × 10^(-3) kg
The fundamental frequency of a guitar string is given by:
f_guitar = (1 / 2L_guitar) * √(Tension / μ)
Where:
Tension is the tension in the string, and
μ (mu) is the linear mass density of the string.
To find μ, we divide the mass of the string by its length:
μ = m / L_guitar = 0.4 × 10^(-3) kg / 0.3 m
Now, substitute the given values into the equation for the fundamental frequency of the guitar string:
f_guitar = (1 / 2L_guitar) * √(Tension / μ)
= (1 / 2 * 0.3) * √(78 / (0.4 × 10^(-3) / 0.3))
Simplify the equation:
f_guitar = 0.5 * √(78 / (0.4 × 10^(-3) / 0.3))
Now that we've found the fundamental frequency of the guitar string, we can solve for the length of the pipe:
f_pipe = v / (2L_pipe)
Since we want the pipe to have the same fundamental frequency as the guitar string, we set f_pipe equal to f_guitar:
v / (2L_pipe) = 0.5 * √(78 / (0.4 × 10^(-3) / 0.3))
Now we can solve for L_pipe:
L_pipe = v / (2 * f_guitar * 0.5 * √(78 / (0.4 × 10^(-3) / 0.3)))
Substitute the given values and calculate to find the length of the pipe.
To find the length of the pipe needed to play the same fundamental frequency as the guitar string, we can apply the formula for the fundamental frequency of a pipe that is open at both ends. The formula is given by:
f = (v / 2L)
Where:
f is the frequency
v is the speed of sound in air
L is the length of the pipe
To determine L, we need to find the frequency of the guitar string first. The fundamental frequency of a string is given by:
f = (v / 2L)
Where:
f is the fundamental frequency
v is the speed of waves on the string
L is the length of the string
Given:
Length of the guitar string (L_string) = 30 cm = 0.3 m
Tension on the guitar string (T) = 78 N
Mass of the guitar string (m_string) = 0.4 g = 0.0004 kg
Speed of sound in air (v) = 340 m/s
To find the speed of waves on the string (v_string), we can use the formula:
v_string = sqrt(T / (m_string / L_string))
Substituting the given values:
v_string = sqrt(78 N / (0.0004 kg / 0.3 m))
v_string = sqrt(78 N * 0.3 m / 0.0004 kg)
v_string = sqrt(58500 N*m / 0.0004 kg)
v_string ≈ 7661.7 m/s
Now, substituting the known values into the formula for the fundamental frequency of the string:
f_string = v_string / (2 * L_string)
f_string = 7661.7 m/s / (2 * 0.3 m)
f_string ≈ 12769.5 Hz
Now, substituting the known values into the formula for the frequency of the pipe:
f_pipe = v / (2 * L_pipe)
Since we want the pipe to have the same frequency as the guitar string:
f_pipe = f_string
v / (2 * L_pipe) = v_string / (2 * L_string)
Cross multiplying:
v * L_string = v_string * L_pipe
L_pipe = (v * L_string) / v_string
Substituting the given values:
L_pipe = (340 m/s * 0.3 m) / 7661.7 m/s
L_pipe ≈ 0.013 m = 1.3 cm
Therefore, the pipe, open at both ends, would need to have a length of approximately 1.3 cm to play the same fundamental frequency as the guitar string.