Problem 40 - Physics

Question

The density of ice is \(917 \mathrm{~kg} / \mathrm{m}^{3},\) and the density of sea water is \(1025 \mathrm{~kg} / \mathrm{m}^{3}\). A swimming polar bear climbs onto a piece of floating ice that has a volume of \(5.2 \mathrm{~m}^{3}\). What is the weight of the heaviest bear that the ice can support without sinking completely beneath the water?

Step-by-Step Solution

Answer

The ice can support a polar bear weighing up to 554.68 kg.

Step 1: Understand the Problem

We need to calculate the maximum weight a piece of ice can support while still floating. The key is knowing how buoyancy works: the weight of the displaced water must equal the total weight of the ice and the bear.

Step 2: Calculate the Weight of the Ice

The mass of the ice is the product of its volume and density: \( \text{mass of ice} = \text{volume} \times \text{density} = 5.2 \mathrm{~m}^3 \times 917 \mathrm{~kg/m}^3 \). Thus, the mass of the ice is \(4768.4 \mathrm{~kg}\). The weight of the ice is \( \text{mass} \times g = 4768.4 \mathrm{~kg} \times 9.81 \mathrm{~m/s}^2 \), which is approximately \(46817.004 \mathrm{~N}\).

Step 3: Calculate the Displacement of Water

The volume of water displaced is equal to the volume of the ice, which is \(5.2 \mathrm{~m}^3\). The mass of the displaced water is \(5.2 \mathrm{~m}^3 \times 1025 \mathrm{~kg/m}^3 = 5330 \mathrm{~kg}\). Hence the weight of the displaced water is \(5330 \mathrm{~kg} \times 9.81 \mathrm{~m/s}^2 = 52257.3 \mathrm{~N}\).

Step 4: Determine the Weight of the Bear

Since for the ice to just float without sinking, the total weight (ice plus bear) should equal the weight of the displaced water. So, \(\text{weight of bear} = 52257.3 \mathrm{~N} - 46817.004 \mathrm{~N} = 5440.296 \mathrm{~N}\).

Step 5: Conclusion

The heaviest bear that the ice can support without sinking is 5440.296 N. This means the bear can weigh up to \(\frac{5440.296 \mathrm{~N}}{9.81 \mathrm{~m/s}^2} \approx 554.68 \mathrm{~kg}\).

Key Concepts

Density: The Buoyancy Game Changer

Density is a fundamental concept in buoyancy. It's the mass per unit volume of a substance. In our scenario, we have ice and seawater, each with their own densities. Density determines how these materials interact in water.
For ice, the density is lower than seawater's at 917 kg/m³ compared to 1025 kg/m³. This difference is crucial. The lower density of ice allows it to float.
It means that per volume unit, ice is lighter than seawater.
Importantly, if a substance has a density lower than the fluid it's placed in, it will float. That’s why icebergs float! Understanding this concept helps explain why a polar bear can climb onto floating ice without it sinking immediately—because the ice is less dense than the water around it.

Weight: Balancing Forces in Water

Weight is the force exerted by gravity on an object and is calculated by multiplying the mass of the object by the gravitational acceleration (9.81 m/s² on Earth). In buoyancy problems, the weight of both the floating object and fluid displaced plays a critical role.
Using our example, the ice weighing 46817.004 N already pushes against the force exerted by the water below.

• The weight of the ice is calculated using its mass, derived from its volume and density.
• Adding the bear's weight means increasing the load on the ice.
• This total mustn't exceed the upward buoyant force for the ice and bear to stay afloat.

So, the heaviest bear that the ice can support safely is determined by finding the balance between these forces.

Displacement: Supporting Weight in Water

Displacement is key to understanding buoyancy. It refers to the amount of water an object pushes aside when it is submerged or partially submerged in a fluid. When the ice floats, its displacement creates an upward force.
This force, known as the buoyant force, must equal the weight of the water displaced by the submerged part of the ice.

• We determine the displaced water's weight by multiplying its volume by its density and gravitational acceleration.
• For the bear to stay on the ice without sinking, the ice's displacement must equal or exceed the combined weight of the bear and ice.

Therefore, the water's displacement can support up to 554.68 kg of bear weight, ensuring the ice and bear do not submerge completely.

Floating: How Ice Bears the Load

Floating is the concept of staying on the surface of a liquid without sinking. In essence, an object floats when its upward buoyant force equals its weight. This principle explains why our ice and bear scenario works. The density differential allows the ice to float when unloaded.
As the bear climbs onto the ice, its weight adds to the system, increasing downward forces.

• Buoyant forces react by pushing upward due to water displacement.
• The ice continues to float, assuming the total force (weight of ice plus bear) is equal to or less than the buoyant force.

Thus, for the heaviest bear of about 554.68 kg, the ice can still float because the buoyant force matches the combined weight, maintaining the delicate balance that keeps them from sinking entirely.