A 6.0- m-long string has a mass of 12.5 g. Transverse waves propagate along the string with a speed of 189.0 m/s. One end of the string is forced to oscillate at 120.0Hz with an amplitude of 0.470 cm. What power is transmitted along the string?
To calculate the power transmitted along the string, we need to use the formula:
Power = (1/2) * μ * ν * A^2 * v
Where:
- Power is the power transmitted along the string
- μ is the linear mass density of the string
- ν is the speed of the wave on the string
- A is the amplitude of the oscillation
- v is the frequency of the oscillation
First, let's calculate the linear mass density (μ) of the string. The linear mass density is defined as the mass per unit length. To find μ:
μ = mass / length
Given:
- Mass of the string = 12.5 g
- Length of the string = 6.0 m
μ = 12.5 g / 6.0 m
To convert grams to kilograms, divide by 1000:
μ = 0.0125 kg / 6.0 m
Next, let's calculate the power using the given values:
ν = 189.0 m/s (given)
A = 0.470 cm = 0.0047 m (convert cm to m)
v = 120.0 Hz (given)
Power = (1/2) * μ * ν * A^2 * v
Substituting the values:
Power = (1/2) * 0.0125 kg / 6.0 m * 189.0 m/s * (0.0047 m)^2 * 120.0 Hz
Calculate the expression in parentheses:
(0.0047 m)^2 = 0.00002209 m^2
Power = (1/2) * 0.0125 kg / 6.0 m * 189.0 m/s * 0.00002209 m^2 * 120.0 Hz
Calculate the multiplication:
(1/2) * 0.0125 kg / 6.0 m * 189.0 m/s * 0.00002209 m^2 * 120.0 Hz ≈ 0.006913475 W
So, the power transmitted along the string is approximately 0.0069 W.