# Homework Help: Buoyant forces

1. Jul 8, 2010

### mandy9008

1. The problem statement, all variables and given/known data
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.)

3. 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

2. Jul 8, 2010

### Staff: Mentor

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

3. Jul 8, 2010

### mandy9008

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?

4. Jul 8, 2010

### Staff: Mentor

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.)

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

5. Jul 8, 2010

### 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

6. Jul 8, 2010

### Staff: Mentor

That's Archimedes's Principle--B is the buoyant force.
You're using the wrong volume. What's the volume of displaced fluid?

7. Jul 8, 2010

### mandy9008

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?

8. Jul 8, 2010

### Staff: Mentor

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

9. Jul 8, 2010

### mandy9008

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

ρobject / ρfluid = Vfluid / Vobject

10. Jul 8, 2010

### mandy9008

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

11. Jul 8, 2010

### 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?

12. Jul 8, 2010

### Staff: Mentor

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

13. Jul 8, 2010

### mandy9008

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