A string with a length of 2.5m resonates in five loops. The string linear density is 0.05 kg/m and the suspended mass is 0.5 kg.
What is the wavelength and wave speed
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To find the wavelength and wave speed, we need to use the formula for the wave speed on a string:
v = √(F/μ)
where:
v = wave speed
F = tension in the string
μ = linear density of the string
To find the tension (F) in the string, we can use the equation for the resonant frequency of a string:
f = 1/2L √(F/μ)
where:
f = frequency of vibration
L = length of the string
Given that the string resonates in five loops, we can determine the length of the string as follows:
Length of string = (2.5m) / 5 = 0.5m
Now we can substitute the values into the equation for the frequency:
f = 1/2 * 0.5m * √(F / 0.05 kg/m)
Simplifying, we get:
f = √(F/0.05 kg/m)
Now, let's solve for F:
F = f^2 * 0.05 kg/m
Given that the suspended mass is 0.5 kg, we know that F = mg:
F = (0.5 kg) * (9.8 m/s^2)
Now, substituting this value into the equation for F, we get:
(0.5 kg) * (9.8 m/s^2) = f^2 * 0.05 kg/m
Simplifying, we find:
f^2 = (0.5 kg * 9.8 m/s^2) / 0.05 kg/m
Solving for f:
f = √((0.5 kg * 9.8 m/s^2) / 0.05 kg/m)
Now that we have the frequency, we can use it to find the wave speed (v) by substituting it into the formula:
v = √(F/μ)
v = √(((0.5 kg * 9.8 m/s^2) / 0.05 kg/m) / 0.05 kg/m)
Simplifying, we get:
v = √((0.5 kg * 9.8 m/s^2) / 0.0025 kg/m^2)
Now, we can calculate the wavelength (λ) using the formula:
λ = v/f
λ = (√((0.5 kg * 9.8 m/s^2) / 0.0025 kg/m^2)) / f
Substituting the values, we can find the wavelength and wave speed.