When disturbed, a tank of water of length L and water depth H is observed to slo

Question

When disturbed, a tank of water of length L and water depth H is observed to slosh back and forth at a frequency ohm. The frequency is known to depend on the acceleration of gravity g. Form two dimensionless quantities from the four given quantities. Only one of these should contain the frequency ohm. Suppose the sloshing frequency on Earth is one cycle per second. What would the frequency be on the Moon where the gravity is one-sixth as great (The tank and the amount of water in the tank are the same in both cases.)

Explanation / Answer

>> Using standard dimension system of M,L,T

>> As, dimensions of Length, L are = [ L ]

>> dimensions of Water Depth, H are = [ L ]

>> dimensions of frequency, f are = [ T-1 ]

>> dimensions of Acceleration due to Gravity, g are = [ L T-2 ]

>> So, 1st Dimensionless Quantity = (L/H)

and, 2nd Dimensionless Quantity = (f2L/g)

>> Now, as frequency, depends on acceleration due to gravity, g (given),

and, from 2nd dimensionless quantity, it can be shown that

frequency f, is proportional to (g)1/2

>> So, at moon, where g become (1/6) times of its value at earth,

=> frequency, f at moon will become (1/6)1/2 = 0.41 times its original value at earth

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