Natural frequency of the sloshing water in a swimming pool of a cruiser ship is studied by an experimental set up with a scale of 1/10. If the natural frequency is found 3.5 Hz, then what is the natural frequency for full scale?
time to cross pool = L/v = T
where T is proportional to period or 1/f
so L = k v /f where k is some constant
f = k v/L
now I do not know what you have studied. I hope you know that if wavelength is short compared to depth the speed of the wave is proportional to the square root of its length lambda
v = c L^.5
so
f = k c L^.5 / L
or
f = constant /sqrt L
3.5 = constant /sqrt 1 if L = 1
f = = constant / sqrt (10) if L = 10
3.5/f = sqrt 10/1
f = 3.5 / sqrt 10
To find the natural frequency for full scale, you need to understand the relationship between the scale and the natural frequency. In this case, the experimental set-up has a scale of 1/10, which means that the dimensions of the set-up are one-tenth the size of the actual swimming pool.
The natural frequency of a system is inversely proportional to the square root of the scale factor. Therefore, you can use the following formula to calculate the natural frequency for the full-scale swimming pool:
Frequency(full scale) = Frequency(experimental) * sqrt(Scale factor)
Let's plug in the values:
Frequency(full scale) = 3.5 Hz * sqrt(1/10)
To calculate the square root of 1/10, we can express it as 1/sqrt(10):
Frequency(full scale) = 3.5 Hz * (1/sqrt(10))
Now, let's simplify the expression:
Frequency(full scale) = 3.5 Hz / sqrt(10)
Using a calculator, you can approximate the value:
Frequency(full scale) ≈ 1.108 Hz
Therefore, the natural frequency for the full-scale swimming pool is approximately 1.108 Hz.