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## Homework Statement

A 62.0-kg survivor of a cruise line disaster rests atop a block of Styrofoam insulation, using it as a raft. The Styrofoam has dimensions 2.00 m X 2.00 m X 0.0900 m. The bottom 0.024 m of the raft is submerged.

a) Draw a free-body diagram of the system consisting of the survivor and raft.

b) Write Newton's second law for the system in one dimension. (Use B for buoyancy, w for the weight of the survivor, and w

_{r}for the weight of the raft. Set a = 0.)

c) Calculate the numeric value for B. (Seawater density = 1025 kg/m

^{3})

d) Calculate weight w

_{r}of the Styrofoam.

e) What is the density of the Styrofoam?

f) What is the maximum buoyant force, corresponding to the raft being submerged up to its top surface?

g) What total mass of the survivors can the raft support?

## Homework Equations

[tex]\rho[/tex] = [tex]\frac{M}{V}[/tex]

P = [tex]\frac{F}{A}[/tex]

V = lwh

P = P

_{0}+ [tex]\rho[/tex]gh

[tex]\frac{\rho

_{obj}}{\rho

_{fluid}}[/tex] = [tex]\frac{V

_{fluid}}{V

_{obj}}[/tex]

## The Attempt at a Solution

a) I have a free-body diagram including:

- Force of Gravity (downwards)
- Buoyancy (upwards)

Must I include the Normal force as well?

b) I have:

B = (w + w

_{r}) = 0

B = w + w

_{r}= mg + m

_{r}g

B = m

_{water}g = [tex]\rho[/tex]Vg = [tex]\rho[/tex]lwhg

V

_{raft}= (0.024 m)(2.00 m)(2.00 m) = 0.096 m^3

g(m + m

_{r}) = [tex]\rho[/tex]

_{water}V

_{raft}g

The g's cancel.

c) 62 kg + m

_{raft}= (1025 kg/m^3)(0.096 m^3)

m

_{raft}= 36.4 kg

But based on my equations, 98.4 would also equal the buoyancy. I think there's an error in my attempt to write Newton's second law for the system in (b).

Any guidance would be appreciated.

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