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13. A barge floating in fresh water (Q= 1000 kg/m3) is shaped like a hollow rectangular...

13. A barge floating in fresh water (Q= 1000 kg/m³) is shaped like a hollow rectangular prism with base area A=550 m²and height H= 2.0 m. When empty the bottom of the barge is located H0= 0.45 m below the surface of the water. When fully loaded with coal the bottom of the barge is located H1= 8 m below the surface.

Select the correct free-body diagram for the barge when it is empty.
Write an equation for the buoyant force on the empty barge in terms of the known data.
Determine the mass of the barge in kilograms.
Which free body diagram represents the barge when it is loaded with coal?
Find the mass of the coal in terms of the given data.
Find the mass of the coal in kilograms.
How far would the barge be submerged (in meters) in mc,2= 350000 kg of coal had been placed in the empty barge?

Solutions

Expert Solution

Ho = height of the empty barge below the water surface

Q = density of fresh water

A = area of the base of the barge

Volume of water displaced is given as

V = A Ho

Fb = force of buoyancy on the empty barge

force of buoyancy on the empty barge is given as

Fb = Q V g

Fb = Q (A Ho) g

Fb = Q A Ho g

M = mass of the barge

Using equilibrium of force along the vertical direction , we have

Fb = Fg

Q A Ho g = Mg

(1000) (550) (0.45) = M

M = 247500 kg

m = mass of the coal

H = height of the barge inside water

Q = density of fresh water

A = area of the base of the barge

Volume of water displaced is given as

V = A H

Fb = force of buoyancy on the loaded barge

force of buoyancy on the loaded barge is given as

Fb = Fg + Fg'

Q A H g = Q A Ho g+ mg

m = (Q A) (H - Ho)

m = ((1000) (550)) (2 - 0.45)

m = 852500

m = (Q A) (H1 - Ho)

350000 = ((1000) (550)) (H1 - 0.45)

H1 = 1.086 m

genius_generous answered 1 year ago

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