A rectangular block of ice 8 m on each side and 1.5 m thick floats in sea water. The density of the sea water is 1025 kg/m3. The density of ice is 917 kg/m3.
a) How high does the top of the ice block float above the water level?
b) How many penguins of mass 23 kg each can stand on the ice block before they get their feet wet?
To answer these questions, we need to consider the principles of buoyancy and Archimedes' principle.
a) To determine how high the ice block floats above the water level, we can calculate the submerged volume of the block.
First, let's find the volume of the ice block:
Volume = Length x Width x Height
Volume = 8 m x 8 m x 1.5 m
Volume = 96 m^3
Since ice density is 917 kg/m^3, we can calculate the mass of the ice block:
Mass = Density x Volume
Mass = 917 kg/m^3 x 96 m^3
Mass = 88,032 kg
According to Archimedes' principle, the buoyant force acting on a submerged object is equal to the weight of the fluid displaced by the object. Therefore, the buoyant force acting on the ice block is equal to the weight of the water displaced.
Buoyant Force = Density of Water x Volume of Displaced Water x Gravity
Buoyant Force = 1025 kg/m^3 x 96 m^3 x 9.8 m/s^2
Buoyant Force = 97,843.2 N
The weight of the ice block is equal to its mass multiplied by the acceleration due to gravity:
Weight of Ice Block = Mass x Gravity
Weight of Ice Block = 88,032 kg x 9.8 m/s^2
Weight of Ice Block = 862,257.6 N
Since the ice block floats, the buoyant force is equal to the weight of the ice block:
Buoyant Force = Weight of Ice Block
97,843.2 N = 862,257.6 N
Now, let's find the submerged volume of the ice block:
Buoyant Force = Density of Water x Volume of Displaced Water x Gravity
97,843.2 N = 1025 kg/m^3 x Volume of Displaced Water x 9.8 m/s^2
Solving for the volume of displaced water:
Volume of Displaced Water = 97,843.2 N / (1025 kg/m^3 x 9.8 m/s^2)
Volume of Displaced Water = 9.73 m^3
Since the ice block has the same volume as the displaced water, the volume of the part of the ice block that is above the water level is equal to the total volume of the ice block minus the volume of the displaced water:
Volume above Water Level = Volume of Ice Block - Volume of Displaced Water
Volume above Water Level = 96 m^3 - 9.73 m^3
Volume above Water Level = 86.27 m^3
To find the height, we divide the volume above the water level by the length and width of the ice block:
Height = Volume above Water Level / (Length x Width)
Height = 86.27 m^3 / (8 m x 8 m)
Height = 1.07 m
Therefore, the top of the ice block floats approximately 1.07 meters above the water level.
b) To determine how many penguins of mass 23 kg each can stand on the ice block before their feet get wet, we need to consider the additional weight that the ice block can support without sinking.
The net force acting on the ice block must be zero for it to remain floating. Therefore, the buoyant force must be equal to the weight of the ice block plus the weight of the penguins.
Let's denote the number of penguins by "n." The total weight of the ice block and penguins is then:
Total Weight = Weight of Ice Block + (Mass of Penguin x n)
To remain floating:
Buoyant Force = Total Weight
Using the same equation from part a, we can calculate the volume of displaced water:
Volume of Displaced Water = Buoyant Force / (Density of Water x Gravity)
Volume of Displaced Water = 97,843.2 N / (1025 kg/m^3 x 9.8 m/s^2)
Volume of Displaced Water = 9.73 m^3
Now, we can solve for the number of penguins:
Buoyant Force = Weight of Ice Block + (Mass of Penguin x n)
Rearranging the equation:
Mass of Penguin x n = Buoyant Force - Weight of Ice Block
Mass of Penguin x n = 97,843.2 N - 862,257.6 N
Mass of Penguin x n = -764,414.4 N
Since the mass of a penguin is 23 kg, we can solve for n:
23 kg x n = -764,414.4 N
Solving for n:
n = -764,414.4 N / 23 kg
n ≈ -33,226.3
Based on the negative value obtained for n, it indicates that the buoyant force is not able to support the additional weight of the penguins, and the ice block would sink if any penguins step on it.
Hence, no penguins should stand on the ice block if they want to stay dry.