The speed of a wave in a string is given by v = sqrt (FT/m), where FT is the tension in the string and m = mass / length of the string.
A 2.00 m long string has a mass of 28.50 g. A 1024 g mass is attached to the string and hung over a pulley (see illustration from one of the team problems). The end of the string is then vibrated at a frequency of 134 Hz. Find the wavelength for the wave generated. Give your answer in centimeters (cm) and with 3 significant figures.
To find the wavelength of the wave generated by the vibrating string, we can use the formula:
v = sqrt(FT/m)
Given that the length of the string is 2.00 m and the mass is 28.50 g, we can calculate the mass per unit length (m) of the string:
m = mass / length = 28.50 g / 2.00 m = 14.25 g/m
Next, we need to find the tension in the string (FT). Since a 1024 g mass is attached to the string and hung over a pulley, the tension in the string is equal to the gravitational force acting on the mass:
FT = mg = 1024 g = 1024 * 9.8 N (converting g to N)
Now we have all the values needed to calculate the speed of the wave (v). Plugging in the values:
v = sqrt(FT/m) = sqrt(1024 * 9.8 / 14.25) = sqrt(7025.92) ≈ 83.79 m/s
The speed of the wave is approximately 83.79 m/s.
The speed of a wave (v) is related to its frequency (f) and wavelength (λ) by the equation v = fλ. Rearranging this equation, we can solve for the wavelength:
wavelength = v / frequency = 83.79 m/s / 134 Hz = 0.625 m ≈ 62.5 cm
The wavelength of the wave generated by the vibrating string is approximately 62.5 cm, rounded to three significant figures.