To calculate the amplitude, frequency, velocity, and wavelength of the transverse wave described by the equation y = 5sin(pi(0.02x-4.00t)), we can compare it to the general equation for a transverse wave:
y = A * sin(2π/λ * (x/T - t))
where:
- y is the displacement of the wave (in cm or any other unit of distance)
- A is the amplitude of the wave (in cm or any other unit of distance)
- λ is the wavelength of the wave (in cm or any other unit of distance)
- x is the position coordinate along the direction of propagation (in cm or any other unit of distance)
- T is the period of the wave (in seconds)
- t is the time coordinate (in seconds)
Comparing the given equation y = 5sin(pi(0.02x-4.00t)) to the general equation, we can deduce the following values:
Amplitude (A): The amplitude is given directly as 5 cm.
Frequency (f): Frequency is the reciprocal of the period, so f = 1/T. However, in the given equation, the time variable t is in seconds, so we need to convert the coefficient of t to radians per second. The coefficient of t in the equation is 4.00, which represents the angular frequency ω = 2πf = 4.00 rad/s. Therefore, the frequency can be calculated as f = ω / (2π) = 4.00 / (2π) ≈ 0.637 Hz.
Velocity (v): The velocity of the wave is given by the product of the angular frequency and the wavelength, v = ωλ. In the given equation, the coefficient of x is 0.02, which represents the wave number k = 2π/λ = 0.02. Rearranging the equation, we have λ = 2π/k = 2π/0.02 ≈ 314.16 cm. Now we can calculate the velocity using v = ωλ = 4.00 rad/s * 314.16 cm ≈ 1256.64 cm/s.
Wavelength (λ): We have already calculated the wavelength as λ ≈ 314.16 cm.
To summarize:
- Amplitude (A) = 5 cm
- Frequency (f) ≈ 0.637 Hz
- Velocity (v) ≈ 1256.64 cm/s
- Wavelength (λ) ≈ 314.16 cm