An object of mass M is used to provide tension in a 4.5 m long string that has a mass of 0.252 kg. A standing wave that has a wavelength equal to 1.5 m is produced by a source that vibrates at 30 Hz. Calculate the value of M
To calculate the value of M, we can use the formula for the speed of a wave:
v = λ * f
Where:
v is the velocity of the wave
λ is the wavelength of the wave
f is the frequency of the wave
We can rearrange the formula to solve for the velocity:
v = λ * f
v = 1.5 m * 30 Hz
v = 45 m/s
Since the string is under tension produced by the object of mass M, we can use the formula for the velocity of a wave in a string under tension:
v = √(T/μ)
Where:
v is the velocity of the wave
T is the tension in the string
μ is the mass per unit length of the string, which can be calculated as the mass of the string divided by its length:
μ = mass / length
μ = 0.252 kg / 4.5 m
μ = 0.056 kg/m
Now we can substitute the values into the formula and solve for the tension (T):
45 m/s = √(T / 0.056 kg/m)
Squaring both sides:
(45 m/s)^2 = T / 0.056 kg/m
2025 m^2/s^2 = T / 0.056 kg/m
T = 2025 m^2/s^2 * 0.056 kg/m
T = 113.4 N
Therefore, the tension in the string is 113.4 N, which is equal to the weight of the object hanging on the string. So, the value of M, the mass of the object, is 113.4 N divided by the acceleration due to gravity (9.8 m/s^2):
M = 113.4 N / 9.8 m/s^2
M ≈ 11.6 kg