A large rectangular raft (density 408 kg/m3) is floating on a lake. The surface

Question

A large rectangular raft (density 408 kg/m3) is floating on a lake. The surface area of the top of the raft is 7.4 m2 and its volume is 2.44 m3. The density of the water is 1011 kg/m3. (a) Calculate the height h of the portion of the raft that is above the surrounding water. (b) Calculate the magnitude of the buoyant force on the raft and state its direction. (c) If the average mass of a person is 77 kg, calculate the maximum number of people that can be on the raft without the raft sinking below the surface of the water. (Assume that people are evenly distributed on the raft.)

Explanation / Answer

Mass of Raft = density x volume = 408 kg/m^3 x 2.44 m^3 = 995.52 kg

Height of Raft = Volume/Area = 2.44 m^3 / 7.4 m^2 = 0.329 m

a) For raft to float,

Weight of raft = Buoyant force on raft

mg = density of water x Volume displaced x g

--> m = density of water x Volume displaced (Lets assume that h metres portion of raft is below the water)

--> Volume displaced = Area x h = 7.4 x h

--> m = density of water x 7.4 x h

--> h = m / (density of water * 7.4) = 995.52 / (1011*7.4) =0.133 m below water

--> Portion of raft above water = 0.329-0.133 = 0.196 m = 19.6 cm

b) Buoyant force on raft is in vertically upward direction and its magnitude is equal to the weight of the raft = 995.52 x 9.8 N
Buoyant force = 9756.096 N

c) Lets assume that n people can be there on raft without causing it to sink.
So, maximum buoyant force = total weight
Maximum buoyant force means full raft is in water
density of water x volume of raft X g = (995.52 + 77n) x g
1011 x 2.44 = (995.52 + 77n)
which gives n = 19.1
Thus 19 people can safely be on the raft.

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