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 for the wave generated in the given scenario, we can use the formula:
v = sqrt(FT/m)
where v represents the speed of the wave, FT represents the tension in the string, and m represents the mass per unit length of the string.
Given:
Length of the string (L) = 2.00 m
Mass of the string (m) = 28.50 g = 0.0285 kg
Mass attached to the string (M) = 1024 g = 1.024 kg
Frequency of vibration (f) = 134 Hz
First, let's calculate the tension in the string (FT). The tension can be calculated using the equation:
FT = Mg
where M represents the mass attached to the string and g represents the acceleration due to gravity. Let's assume g is approximately 9.81 m/s^2.
FT = 1.024 kg * 9.81 m/s^2
FT = 10.0449 N
Next, we need to calculate the total mass of the string (M_total) by adding the mass of the string (m) and the mass attached to it (M).
M_total = m + M
M_total = 0.0285 kg + 1.024 kg
M_total = 1.0525 kg
Now, we can calculate the speed of the wave (v) using the formula:
v = sqrt(FT / m)
v = sqrt(10.0449 N / 1.0525 kg)
v = 3.0207 m/s (rounded to 4 decimal places)
The speed of the wave is approximately 3.0207 m/s.
Lastly, to find the wavelength (λ), we can use the equation:
λ = v / f
λ = 3.0207 m/s / 134 Hz
λ = 0.02255 m
Now, let's convert the wavelength from meters to centimeters. There are 100 centimeters in 1 meter, so:
λ = 0.02255 m * 100 cm/m
λ = 2.255 cm
Therefore, the wavelength for the wave generated is approximately 2.255 cm (rounded to 3 significant figures).