Calculating Buoyancy in a Disaster Survivor Scenario

[mandy9008]

Homework Statement

A 63 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 multiplied by 2.00 m multiplied by 0.04 m. The bottom 0.026 m of the raft is submerged.

Write Newton's second law for the system in one dimension, using B for buoyancy, w for the weight of the survivor, and wr for the weight of the raft. (Set a = 0. Use w for w, and w_r for I>wr as needed)

Calculate the numeric value for the buoyancy, B. (Seawater has density 1025 kg/m3.)

The Attempt at a Solution

Fy: B-w-w_r

I need to calculate the weight of the raft. the only thing that I could think to do, given the values in the problem, was to use density=m/v
1025 kg/m3 = m/0.16m3
m=164kg (this seems way to high to be the mass of Styrofoam)

(I got the volume by multiplying the dimensions of the raft)

B= 63kg-164kg
B=101 kg

 

[Doc Al]

I need to calculate the weight of the raft.

Why? I thought you just needed to calculate the buoyant force?

 

[mandy9008]

Why? I thought you just needed to calculate the buoyant force?

I do, but I figured that, since I was asked to provide the equation for Fy (B-w-w_r), I would need to use that. Is this not the case?

I know:
V = 0.16 m3
ΔV = 0.026 m
m = 63 kg

In order to use the bulk modulus, I need to know P. Do i just use P for water?

 

[Doc Al]

I do, but I figured that, since I was asked to provide the equation for Fy (B-w-w_r), I would need to use that. Is this not the case?

No, you don't need that to solve for the buoyant force. (But you'll need it to solve for the mass of the raft, if they ask that later.)

In order to use the bulk modulus, I need to know P. Do i just use P for water?

B is the buoyant force, not the bulk modulus. You are given the density of seawater. You'll need that and Archimedes' principle.

 

[mandy9008]

i just found another equation for B that my prof gave me. It is B=ρVg.
B=(1025 kg/m3)(0.16m3)(9.8m/s2)
B=1607.2N

this is wrong though

 

[Doc Al]

i just found another equation for B that my prof gave me. It is B=ρVg.

That's Archimedes's Principle--B is the buoyant force.

B=(1025 kg/m3)(0.16m3)(9.8m/s2)

You're using the wrong volume. What's the volume of displaced fluid?

 

[mandy9008]

That's Archimedes's Principle--B is the buoyant force.

You're using the wrong volume. What's the volume of displaced fluid?

the volume I used was the volume of the block. when it is put in the water, 0.026m of it is submerged. does this mean that i need to subtract this value from the volume of the block?

 

[Doc Al]

the volume I used was the volume of the block. when it is put in the water, 0.026m of it is submerged. does this mean that i need to subtract this value from the volume of the block?

0.026m is the depth which is submerged. What's the volume that's submerged?

 

[mandy9008]

the only equation in my textbook i can find shows this:

ρ_object / ρ_fluid = V_fluid / V_object

 

[mandy9008]

i just figured it out.
2m x 2m x 0.026m = 0.104m3

 

[mandy9008]

Using the value of B and the weight w of the survivor, calculate the weight wr of the Styrofoam.
B=w + w_r
1044.68N = 617.4N + w_r
w_r = 427.3 N


What is the density of the Styrofoam?
ρ=m/v
ρ=(43.6 kg) / 0.16 m3
ρ=272.5 kg/m3


What is the maximum buoyant force, corresponding to the raft being submerged up to its top surface?
B=ρvg
B=(1025 kg/m3)(0.16m3)(9.8 m/s2)
B=1607.2 N


What total mass of survivors can the raft support?
which equation can I use for this?

 

[Doc Al]

What total mass of survivors can the raft support?
which equation can I use for this?

Use the same force equation. Only you have a new value for B and you're solving for w.

 

[mandy9008]

okay, i was just using the wrong buoyancy force. Thanks! :)