It is required to determine the power P ([P] = M L^2 T^-3) that must be applied

Question

It is required to determine the power P ([P] = M L^2 T^-3) that must be applied to keep a ship of length l ([l] = L) moving at a constant speed V ([V] = LT^-1) in water. It is known that P depends on the density of water rho ([rho] = ML^-3), the acceleration due to gravity g ([g] = LT^-2). and the kinematic viscosity of water v ([v] = L^2T^-1), as well as l and V. Use dimensional reduction to show that P/rho l^2 V^3 = G (Fr, Re) where G is an unknown function, Fr is the Froude number, and Re is Reynold's number, with Fr = V/Squareroot gl, Re = Vl/v A streamlined ship is to be built where the effect of viscosity can be neglected, so that the result of part (a) simplifies to P/rho l^2 V^3 = H (V/Squareroot gl) where H is some unknown function. The ship is to be 100 m in length, and is to have a cruising speed of 36 km hr^-1. Engineers build a geometrically similar 1/50 th scale model of the proposed ship. Assuming complete dynamic similarity between the ship and the model, at what speed should the engineers run the model in order to mimic the behavior of the ship at cruising speed? When the engineers run the model at this speed, they find that a power of 0.04 horsepower must be applied to maintain it. Based on this measurement, estimate the power required to maintain the ship at cruising speed.

Explanation / Answer

1 a)

Given

Length of ship = l (L)
Speed = V (LT^-1)

Density of water = (ML^-3)

Acceleration due to gravity = g (LT^-2)

To Prove that

P/l^2V^3 = G(Fr, Re)

Fr = Froude Number, Re = Reynold's No.

LHS = P/l^2V^3 =

ML^2/T^-3/ (M/L^3) (L^2) (L)^3

L^3/ T^-3(L)^3

= 1/T^-3 = T^3

RHS = Since RHS depends upon

Fr = Froude Number & Re = Reynold's No.

Hence their are constant values

Hence dimesional formula of G = T^3

Since LHS = RHS

Hence the relation is P/l^2V^3 = G(Fr, Re) proved.

Note:- Since 1(a) part is itself an indepndent question so only this question has been solved.

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