Question

5. (a) In Grand Bahama canals were built to transport logs from inland to the sea. A canal is full to depth H, find an expression for the force per unit length exerted on the canal's wall by the water, if we assume constant density. [6] (b) The geology of The Bahamas is such that the fresh water sits on more saline water. If we assume a linear change of density with depth such that where is the density of the fresh water, find the new force per unit length exerted on the walls of the canal. [6]

Answer 1

a)

conside a canal with height H and length L,with water of constant density is flowing through it,

we know that the presser exerted at a depth H is given by P=Hρg

(where ρ is the density,g is the gravitational constant and H is the height)

and the general expression for presser is given by P=F/A (where F is the foece acting,A is the area of cross section)

that is dP=dF/dA ,dF=dP*dA

now presser exerted at a distance x is given by P=xρg

dA=dx*dL

total force acting F=∫dF=∫xρg*dx*dL=Lρg*x2/2|0H=LρgH2/2

so force per unit length =F/L=ρgH2/2

b)

if there is a linear change in density with depth ,then density at a depth x is given by ρ=ρ0x

(every thing will be reamains the same)

presser exerted at a distance x is given by P=xρg=x*ρ0x*g=ρ0x2g

(everything remains the same)

total force acting F=∫dF=∫ρ0x2g*dx*dL=L*ρ0g*X3/3=L*ρ0g*H3/3

force acting per unit length =F/L=ρ0g*H3​​​​​​​/3