Problem 87 - Fundamentals of Physics

Question

What is the minimum area (in square meters) of the top surface of an ice slab \(0.441 \mathrm{~m}\) thick floating on fresh water that will hold up a \(938 \mathrm{~kg}\) automobile? Take the densities of ice and fresh water to be \(917 \mathrm{~kg} / \mathrm{m}^{3}\) and \(998 \mathrm{~kg} / \mathrm{m}^{3},\) respectively.

Step-by-Step Solution

Answer

The minimum area is approximately 25.99 square meters.

Step 1: Understand the Problem

We need to find the minimum area of the top surface of an ice slab that is thick enough to support a car when floating on water. We are given the thickness of the ice and the densities of both the ice and the water.

Step 2: Apply Archimedes' Principle

According to Archimedes' principle, the weight of the water displaced by the floating ice equals the total weight of the ice plus the weight of the car. \[ \text{Buoyant Force} = (938 + \text{mass of ice})g \] where \(g\) is the acceleration due to gravity.

Step 3: Calculate the Mass of the Ice

The volume of the ice is calculated using its thickness and area \(A\). The mass of the ice is thus \(\rho_{ice} \times 0.441 \times A\). Therefore, \(\text{mass of ice} = 917 \times 0.441 \times A\).

Step 4: Setup the Equation for Buoyant Force

The buoyant force is equal to the weight of the water displaced, which is \(\rho_{water} \times V_{displaced} \times g\). Since the ice is floating, the entire volume \(0.441 \times A\) displaces water. So, \(\rho_{water} \times 0.441 \times A \times g = (938 + 917 \times 0.441 \times A)g\).

Step 5: Solve for Area \(A\)

Rearrange the equation: \[998 \times 0.441 \times A = 938 + 917 \times 0.441 \times A\]. Simplifying, \[(998 - 917) \times 0.441 \times A = 938\]. Calculate \(A\): \[81 \times 0.441 \times A = 938\]. \[A = \frac{938}{81 \times 0.441}\].

Step 6: Calculate the Minimum Area

Upon calculation, \[A \approx 25.99\]. Therefore, the minimum area of the top surface of the ice slab that will hold up the car is approximately \(25.99 \text{ m}^2\).

Key Concepts

Buoyant Force

The concept of buoyant force is key to understanding why objects float or sink in a fluid. Archimedes' Principle explains that when an object is immersed in a fluid, it experiences an upward force known as buoyant force. This force is equivalent to the weight of the fluid that the object displaces.
This principle is crucial in solving problems involving floating objects, such as ice slabs supporting an automobile. It helps determine if the upward buoyant force can counterbalance the downward forces due to gravity acting on the object. As a result, understanding buoyant force gives insight into why specific conditions cause objects to float in water, like in this exercise where the ice must properly support the car's weight.

Density of Ice

Density is defined as mass per unit volume, and the density of ice is an essential factor in determining buoyancy. In this exercise, the density of ice is given as 917 kg/m³. This value is slightly less than that of water, making ice less dense, allowing it to float.
When comparing densities, if the object's density is less than that of the fluid, it floats. Otherwise, it will sink. This relationship is vital to know how ice can support additional weight, such as a car. The density helps in calculating the mass of the ice involved, which plays a crucial role in the balance of forces needed for floating.

Fresh Water Buoyancy

Fresh water buoyancy refers to the ability of objects to float in freshwater environments like lakes or rivers. The buoyant force depends on the density of the fluid, which is 998 kg/m³ for fresh water. This density is used to calculate the volume of water displaced when a solid object like an ice slab floats.
When an ice slab floats, it displaces water equal to its volume under the surface. This displaced water generates the necessary buoyant force to counteract the weight of the ice and any additional weight, like a car. Therefore, understanding fresh water buoyancy helps explain the stability and flotation of objects in aquatic settings.

Mass and Volume Calculations

Calculating mass and volume accurately is crucial in buoyancy-related problems. For an ice slab, its mass can be determined by multiplying its density (917 kg/m³) by its volume. The volume of the ice is the product of its thickness (0.441 m) and the area of its top surface.

• Mass of ice = Density of ice × Volume of ice
• Volume of ice = Thickness × Area

These calculations are integral to figuring out the total weight of the ice slab, which includes using the displaced water volume to understand the buoyant force at play. These methods allow for determining the surface area needed for stability and support on various objects like carriages and vehicles.