a stretched string is observed to vibrate with frequency 30Hz in its fundamental mode when the supports are 60cm apart. the amplitude at the antinode is 3cm.the string has a mass of 30kg.calculate the speed of propagation of the wave and the tension
To calculate the speed of propagation of the wave on the string, we can use the formula:
v = f * λ
where v is the speed of the wave, f is the frequency, and λ is the wavelength.
First, we need to find the wavelength. In the fundamental mode of vibration, the wavelength is twice the distance between the supports, so:
λ = 2 * 60 cm = 120 cm = 1.20 m
Next, we need to calculate the speed of the wave:
v = 30 Hz * 1.20 m = 36 m/s
Therefore, the speed of propagation of the wave on the string is 36 m/s.
To calculate the tension in the string, we can use the formula:
T = (μ * v^2) / L
where T is the tension, μ is the linear mass density of the string (mass per unit length), v is the velocity of the wave, and L is the length of the string.
First, let's calculate the linear mass density:
μ = mass / length
= 30 kg / 60 cm
= 0.5 kg/m
Next, we can calculate the tension:
T = (0.5 kg/m * (36 m/s)^2) / 0.60 m
= (0.5 kg/m * 1296 m^2/s^2) / 0.60 m
= 648 N
Therefore, the tension in the string is 648 N.