Question

A 63.0-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 ✕ 2.00 m ✕ 0.0895 m. The bottom 0.023 m of the raft is submerged.

(a) Draw a force diagram of the system consisting of the survivor and raft.

(b) 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. Solve for Fy, the y-component of the net force. Let upward be the positive y-direction.)

(c) Calculate the numeric value for the buoyancy, B. (Seawater has density 1025 kg/m3. Enter answer to at least the ones digit.)

(d) Using the value of B and the weight w of the survivor, calculate the weight wr 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 survivors can the raft support?

Answer 1

(a) Here is the free body diagram

Notice that the weight of raft and weight of survivor acts down.

wr is weight of raft, w is weight of survivor and B is buoyant force.

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(b)

Using Newton's second law, we have

B - w - w_r = 0

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(c) B = * g * V

where V is volume displaced

B = 1025 * 9.8 * 2 * 2 * 0.023

B = 924.14 N

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(d) using Newton's second law equation found above

w_r = B - w

w is weight of survivor

w_r = 924.14 - (63 * 9.8)

w_r = 306.74 N

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(e)

density = mass / volume

where

m = 306.74 / 9.8 = 31.3 kg

V = 2 * 2 * 0.0895 = 0.358 m³

so,

density = 87.43 kg / m³

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(f) here, we will consider total volume of raft

B = 1025 * 9.8 * 2 * 2 * 0.0895

B = 3596.11 N

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(g) again use Newton's equation

3596.11 - m * 9.8 - 306.74 = 0

3596.11 - 306.74 =  m * 9.8

m = 335.65 kg