A string vibrates in its fundamental mode with a frequency of 200 Hz. The string is 0.64 m long and has a mass of 1.61 g. With what tension must the string be stretched?
v = (T/mu)^0.5
where v is the speed, T is the tension, and mu is the linear density
v = lamda*f ; lambda = 2*L for the fundamental mode, where f is the frequency, and lambda is the wavelength
f = (1/(2*L))*(T/mu)^0.5
1.61 g = 0.00161 kg
mu = 0.00161/0.64
200 = (1/(2*0.64))*(T/mu)^0.5
Plug in mu; use algebra to solve for T
To find the tension in the string, we can use the wave equation:
v = √(T/μ)
Where:
v is the velocity of the wave (which can be calculated using the equation v = f * λ, where f is the frequency and λ is the wavelength).
T is the tension in the string.
μ is the linear mass density of the string (which can be calculated using the equation μ = m/L, where m is the mass of the string and L is the length of the string).
First, let's find the wavelength (λ) of the vibrating string. The fundamental mode of vibration corresponds to a standing wave pattern with one antinode in the middle of the string. In this mode, the wavelength is twice the length of the string:
λ = 2 * L = 2 * 0.64 m = 1.28 m
Next, we can calculate the velocity (v) of the wave:
v = f * λ = 200 Hz * 1.28 m = 256 m/s
Now, we can calculate the linear mass density (μ) of the string:
μ = m / L = 1.61 g / 0.64 m = 2.5156 g/m = 0.0025156 kg/m
Finally, let's calculate the tension (T) in the string using the wave equation:
v = √(T/μ) => T = v^2 * μ = (256 m/s)^2 * 0.0025156 kg/m = 16.6616 N
Therefore, the tension in the string must be approximately 16.66 N.