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 of the ice's top surface is approximately \( 2.14 \, \text{m}^2 \).

Step 1: Understand Archimedes' Principle

Archimedes' Principle states that any floating object displaces its own weight of fluid. To find the minimum area of ice needed to support the car, the buoyant force must equal the weight of the car.

Step 2: Calculate the Weight of the Automobile

The weight of the car, which Archimedes' principle mentions, is the gravitational force on it. Compute it using the formula \( W = mg \). Given \( m = 938 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \), the weight is \( W = 938 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 9192.4 \, \text{N} \).

Step 3: Determine Upwards Buoyant Force Exerted by Ice Slab

For equilibrium, the upwards buoyant force must equal the weight of the car. This force is \( F_b = V \cdot \text{density of water} \cdot g \), where \( V \) is the volume of water displaced by the submerged part of the ice.

Step 4: Calculate Ice Volume Needed to Displace Water Equal to Car's Weight

The volume of displaced water \( V \) can be expressed as \( V = F_b / (\text{density of water} \cdot g) = 9192.4 / (998 \times 9.8) \). This yields \( V \approx 0.945 \, \text{m}^3 \).

Step 5: Relate Ice Volume and Area

The ice volume is related to the cross-sectional area \( A \) and thickness \( t \) by \( V = A \cdot t \). Here, \( t = 0.441 \, \text{m} \). Solve for \( A \) using \( A = V/t = 0.945 / 0.441 \).

Step 6: Compute Minimum Area for Stability

Calculate \( A = \frac{0.945}{0.441} \approx 2.14 \, \text{m}^2 \). This is the minimum area needed for the ice block's top surface to support the automobile while floating stably.

Key Concepts

Buoyant Force

The buoyant force is a fundamental concept of fluid mechanics. It's the upward force that a fluid exerts on an object submerged in it. This force is crucial in understanding why objects float. According to Archimedes' Principle, a floating object displaces its own weight of the fluid it is immersed in. This means that the buoyant force on an object at rest in a fluid equals the weight of the fluid displaced by that object.

• This force allows ships to float despite their massive weight.
• The buoyant force depends on the object's volume within the fluid, not on its weight.
• It ensures that the object doesn't sink as long as it is less dense than the surrounding fluid.

To balance the weight of an object, this force must be sufficient to prevent it from sinking. Understanding buoyant force helps in problems such as finding the size of an ice slab needed to support a car as the floating object.

Floating Object

A floating object is one that remains on the surface of a fluid without fully sinking. The object remains buoyant thanks to the interaction between gravity (pulling it down) and the buoyant force (pushing it up). When an object floats, the weight of the object is exactly balanced by the buoyant force.

• The volume of the object submerged changes until this equilibrium is achieved.
• The key to floating lies in the density of the object compared to the fluid.
• Less dense objects will float higher, partially emerging from the fluid.
• This principle is why ice, which is less dense than water, floats on top of it.

Calculating the characteristics of a floating object involves understanding these principles to ensure the correct balance and support.

Density of Water

The density of water is an essential property when discussing buoyancy. Water's density affects how much of an object needs to be submerged to stay afloat, making it a key component in calculations involving floating objects. Typically, the density of fresh water is about 998 kg/m³.

• This number is used to determine how much water an object displaces when submerged.
• It's crucial for calculating the buoyant force, as seen in the formula: \[ F_b = V \times \text{density of water} \times g \]
• A higher density of the fluid provides a greater buoyant force.

When comparing densities of different materials and fluids, this helps to predict floating and sinking behavior.

Volume Displacement

Volume displacement is a key factor in understanding buoyant forces. It refers to the volume of fluid that a submerged part of an object displaces. When an object is placed in a fluid, it pushes aside the fluid, creating volume displacement. This is directly related to the buoyant force experienced by the object.

• The volume displaced directly impacts the magnitude of the buoyant force.
• For an object to float, it needs to displace a volume of fluid equal to its own weight.
• Using the formula for buoyant force, volume displacement is calculated as: \[ V = \frac{F_b}{\text{density of water} \times g} \]
• This helps to calculate how much of the object needs to be submerged for it to float stably.

Understanding and calculating volume displacement is essential when determining the floating capacity of objects like ice slabs with heavy loads.