A sinusoidal wave on a string is described by the equation y = (0.169 m) sin (0.713 x - 41.9 t), where x and y are in meters and t is in seconds. If the linear mass density of the string is 10.1 g/m ...
a) ... the phase of the wave at x = 2.27 cm and t = 0.175 s. ?
b)... the speed of the wave ?
c)... the wavelength.
d) ... the power transmitted by the wave. ?
To answer these questions, we will need to analyze the given equation for the wave and use relevant formulas. Let's go through each question step by step:
a) To find the phase of the wave at x = 2.27 cm and t = 0.175 s, we use the equation: y = (0.169 m) sin (0.713 x - 41.9 t).
Given:
x = 2.27 cm = 0.0227 m
t = 0.175 s
First, we substitute the given values into the equation:
y = (0.169 m) sin (0.713 * 0.0227 - 41.9 * 0.175)
Now we can calculate the phase of the wave by evaluating the expression inside the sine function.
Phase = 0.713 * 0.0227 - 41.9 * 0.175
Calculate the value using a calculator:
Phase ≈ -0.8448
Therefore, the phase of the wave at x = 2.27 cm and t = 0.175 s is approximately -0.8448 radians.
b) To find the speed of the wave, we need to know the relationship between the angular frequency (ω) and the wave number (k). The speed (v) of the wave is given by the formula: v = ω/k.
In our equation, we can see that ω = 41.9 and k = 0.713. Substituting these values into the formula:
v = 41.9 / 0.713
Calculate the value using a calculator:
v ≈ 58.8122 m/s
Therefore, the speed of the wave is approximately 58.8122 m/s.
c) The wavelength (λ) of a wave can be calculated using the formula: λ = 2π/k.
In our equation, k = 0.713. Substituting this value into the formula:
λ = 2π / 0.713
Calculate the value using a calculator:
λ ≈ 8.827 m
Therefore, the wavelength of the wave is approximately 8.827 m.
d) The power transmitted by a wave can be calculated using the formula: power = (1/2) * ρ * A * v^2, where ρ is the linear mass density, A is the amplitude, and v is the speed.
Given:
ρ = 10.1 g/m = 0.101 kg/m
A = 0.169 m
v = 58.8122 m/s
Substituting these values into the formula:
power = (1/2) * 0.101 * 0.169^2 * 58.8122^2
Calculate the value using a calculator:
power ≈ 0.0344 W
Therefore, the power transmitted by the wave is approximately 0.0344 Watts.