Answer:
a) when the length of the rope is doubled and the angular frequency remains constant: The power increases by factor of two
b) when the amplitude is doubled and the angular frequency is halved: The power is the same
c) when both the wavelength and the amplitude are doubled: The power increases by a factor of 8
d) when both the length of the rope and the wavelength are halved: The power increases by factor of two
Explanation:
For a sinusoidal mechanical wave (Transverse wave), the time-averaged power is the energy associated with a wavelength divided by the period of the wave.
[tex]P_{avg} =\frac{E \lambda}{T}=\frac{1}{2} \mu A^2 \omega^2 \frac{\lambda}{T}[/tex]
where;
A is the Amplitude
ω is the angular frequency
λ is the wavelength
a) when the length of the rope is doubled and the angular frequency remains constant
L = λ/2, λ = 2L
The power increases by factor of two
b) when the amplitude is doubled and the angular frequency is halved
[tex]P_{avg} ={\frac{1}{2} \mu (2A)^2 (\frac{\omega}{2}) ^2 \frac{\lambda}{T} = \frac{1}{2} \mu A^2 \omega^2 \frac{\lambda}{T}[/tex]
The power is the same
c) when both the wavelength and the amplitude are doubled
[tex]P_{avg} =\frac{1}{2} \mu (2A)^2 \omega^2 \frac{(2\lambda)}{T} = \frac{1}{2} \mu (4A^2) \omega^2 \frac{(2\lambda)}{T} = 8 (\frac{1}{2} \mu A^2 \omega^2 \frac{\lambda}{T})[/tex]
The power increases by a factor of 8
d) when both the length of the rope and the wavelength are halved
L = λ/2
when both are halved
L/2 = λ/4, λ = 2L
The power increases by factor of two
