Question

If you have ever heard the term 'the tip of the iceberg' you probably know that the amount of an iceberg you see floating in the ocean is only a small amount of the total volume of the iceberg. Most of the iceberg is submerged. For this problem we will determine the ratio of the submerged iceberg volume to the volume above water. Assume we see a volume of 1000 m^3 of ice above the water. The density of water is 1000 kg/m^3 and the density of Ice is 917 kg/m^3. You may assume the iceberg is a perfect cube as shown. AO & Pice = 917 Volume block lwH Volume displaced e wh ing Swater = 1000 kg / H 13h Question 1: Symbolically give the equation for the forces acting on the block of ice in terms of the density and volume and 'g: Remember since the iceberg is floating, it is in equilibrium and the net acceleration is 0.

Answer 1

Given:

$V_{o}$ = volume of ice above water =

1000m

Pw 1000kg/m

Pice = 917kg/m

h = height of ice inside the water

H = Total height of the ice

1) Let m = mass of the ice and V = total volume of ice

then the volume of ice inside water is $V_{i}=V-V_{o}$

Two forces acting on the ice:

a. force due to gravity = mg $=V\rho_{ice}g$

b. buoyant force  $=V_{i}\rho_{w}g$

Since the ice is floating, the net force on the ice is zero, therefore,

the buoyant force is balancing the weight of the ice

=> buoyant force = weight

$\Rightarrow V_{i}\rho_{w}g = V\rho_{ice}g$------(i)

2) Let the cross-section area be A parallel to the horizontal plane.

then, and

Therefore, using the equation developed in part 1:

$Ah\rho_{w}g = AH\rho_{ice}g$

$\Rightarrow h\rho_{w}= H\rho_{ice}$

$\Rightarrow \frac{h}{H}= \frac{\rho_{ice}}{\rho_{w}}$

  $= \frac{917}{1000}=0.92$ [answer]

3) We know, $V=V_{i}+V_{o}$ [Vi = volume inside water, Vo = volume of ice above water]

Now, using the equation developed in part 1:

$V_{i}\rho_{w}g = V\rho_{ice}g$

$\Rightarrow V_{i}\rho_{w} = (V_{i}+V_{o})\rho_{ice}$

$\Rightarrow V_{i}= \frac{V_{o}\rho_{ice}}{\rho_{w}-\rho_{ice}}$

$\Rightarrow V_{i}= \frac{1000*917}{1000-917}=11048.19m^{3}$[answer]

The total volume of the iceberg is $V=11048.19+1000=12048.19m^{3}$ [answer]