The bottom of a flat-bottom aluminum boat has an area = 4 m2 and the mass of the boat is 65 kg.?

Question

The bottom of a flat-bottom aluminum boat has an area = 4 m2 and the mass of the boat is 65 kg. If two fishermen and their fishing gear with total mass of 375 kg are placed in the boat, how much lower will the boat ride in water? (water density = 1000 kg/m3)

Answer 1

In either case, the boat is at equilibrium, which means the sum of the forces acting on it is 0. Weight is a downward force and is countered by the water's buoyancy force (F_B).

-W + F_B = 0, W = Mg, F_B = P_B * A_b

P_B = buoyancy pressure of water = p_w * g * d

where p_w = density of water
g = acceleration of gravity
d = depth of water that the boat/cargo submerges to

A_b = area of the boat in direct contact with water
M = mass of the boat and any cargo or occupants

Getting back to the above force equation:

F_B = W

p_w * g * d * A_b = Mg

Since g is present on both sides of the equation, it can be canceled out:

p_w * d * A_b = M

You are dealing with two scenarios....the first is when the boat is empty, the second when the boat is occupied. In other words....

(1) p_w * d_b * A_b = M_b and

(2) p_w * d_bf * A_b = M_bf

where

d_b = depth that the empty boat submerges to
d_bf = depth that the boat submerges to when it's "full"
(fisherman and their gear are onboard)
M_b = mass of the boat
M_bf = mass of the boat, fisherman, & cargo

You are interested in finding the quantity (d_bf - d_b), the difference in the submergence that the boat experiences when the fisherman and their gear are present. If you subtract the 1st scenario from the 2nd one, you can isolate this quantity:

p_w * A_b * (d_bf - d_b) = (M_bf - M_b)

d_bf - d_b = (M_bf - M_b)/(p_w * A_b)

= 375 kg/(1000 kg m^-3 * 4 m^2)

= 375 kg/4000 kg m^-1

= 0.09375 meters

The boat submerges an additional 0.09375 m when the fisherman and their gear are inside it.

Answer 2

Flat Bottom Aluminum Boat