The speed of a wave in a string is given by v = Ö(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 15.5 g. A 93 g mass is attached to the string and hung over a pulley. The end of the string is then vibrated at a frequency of 133 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, we can use the equation:
v = sqrt(FT / m)
Given:
Length of the string, L = 2.00 m
Mass of the string, m_s = 15.5 g = 0.0155 kg (converting grams to kilograms)
Mass attached to the string, m = 93 g = 0.093 kg (converting grams to kilograms)
Frequency of vibration, f = 133 Hz
First, let's find the tension in the string (FT). The mass attached to the string is hung over a pulley, causing an additional tension. So, the total tension (FT) in the string can be calculated by:
FT = (m + m_s) * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
FT = (0.093 kg + 0.0155 kg) * 9.8 m/s^2
FT = 1.0735 N
Now, let's calculate the speed of the wave (v) using the formula:
v = sqrt(FT / m)
v = sqrt(1.0735 N / 0.0155 kg)
v ≈ 12.185 m/s
Next, we can find the wavelength (λ) using the formula:
v = λ * f
Rearranging the formula, we have:
λ = v / f
λ = 12.185 m/s / 133 Hz
λ ≈ 0.0917 m (rounded to 4 decimal places)
Finally, let's convert the wavelength from meters to centimeters. Since 1 meter is equal to 100 centimeters, the wavelength in centimeters can be calculated as:
λ_cm = λ * 100
λ_cm = 0.0917 m * 100
λ_cm ≈ 9.17 cm
Therefore, the wavelength for the wave generated is approximately 9.17 cm (rounded to 3 significant figures).
To find the wavelength of the wave generated, we can use the formula:
v = sqrt(FT / m)
First, let's find the tension (FT) in the string.
Tension in the string (FT) can be calculated by considering the weight of the attached mass.
Weight = mass * acceleration due to gravity
Weight = 0.093 kg * 9.8 m/s^2 (converting grams to kilograms)
Weight = 0.9114 N
Since the string is under tension due to this weight, FT = Weight = 0.9114 N
Now, let's find the mass per unit length (m) of the string.
m = mass / length
m = 0.0155 kg / 2.00 m (converting grams to kilograms)
m = 0.00775 kg/m
Next, we can substitute the given values into the formula for the wave speed (v):
v = sqrt(FT / m)
v = sqrt(0.9114 N / 0.00775 kg/m)
v ≈ 44.105 m/s
Since the wave speed (v) is equal to the product of the frequency and wavelength (v = f * λ), we can rearrange the formula to solve for the wavelength (λ):
λ = v / f
λ = 44.105 m/s / 133 Hz
λ ≈ 0.3315 m
Finally, let's convert the wavelength to centimeters:
λ ≈ 0.3315 m * 100 cm/m
λ ≈ 33.15 cm
Therefore, the wavelength of the generated wave is approximately 33.15 cm (with 3 significant figures).