Problem 78 - Fundamentals of Physics

Question

Caught in an avalanche, a skier is fully submerged in flowing snow of density \(96 \mathrm{~kg} / \mathrm{m}^{3}\). Assume that the average density of the skier, clothing, and skiing equipment is \(1020 \mathrm{~kg} / \mathrm{m}^{3} .\) What percentage of the gravitational force on the skier is offset by the buoyant force from the snow?

Step-by-Step Solution

Answer

Approximately 9.41% of the gravitational force is offset by the buoyant force from the snow.

Step 1: Understand the Buoyant Force

The buoyant force on an object submerged in a fluid is given by Archimedes' principle. It states that the buoyant force is equal to the weight of the fluid displaced by the object. First, we need to calculate the buoyant force on the skier using the formula: \( F_b = \rho_{\text{snow}} \cdot V \cdot g \), where \( \rho_{\text{snow}} \) is the density of the snow, \( V \) is the volume of the skier, and \( g \) is the acceleration due to gravity.

Step 2: Calculate the Gravitational Force

The gravitational force on the skier can be calculated using the formula: \( F_g = \rho_{\text{skier}} \cdot V \cdot g \), where \( \rho_{\text{skier}} \) is the density of the skier with clothing and equipment, \( V \) is the volume of the skier, and \( g \) is the acceleration due to gravity.

Step 3: Determine the Percentage Offset

The percentage of the gravitational force that is offset by the buoyant force is given by the ratio \( \frac{F_b}{F_g} \times 100\% \). Substituting the expressions from Steps 1 and 2, we get: \( \frac{\rho_{\text{snow}}}{\rho_{\text{skier}}} \times 100\% \). Substituting the values, we have \( \frac{96}{1020} \times 100\% \).

Step 4: Calculate the Final Answer

Simplify the expression from Step 3 to find the percentage: \( \frac{96}{1020} \approx 0.0941 \). Multiply by 100 to convert to a percentage: \( 0.0941 \times 100 \% \approx 9.41 \% \). Therefore, about 9.41% of the gravitational force on the skier is offset by the buoyant force from the snow.

Key Concepts

Archimedes' Principle

Archimedes' principle is a fundamental concept in fluid mechanics that helps us understand how objects behave in fluids. According to this principle, when an object is submerged in a fluid, it experiences an upward force called the buoyant force. This force is equal to the weight of the fluid that the object displaces.

In simpler terms, if you place something in a liquid or gas, it pushes some amount of that fluid aside. Archimedes' principle tells us that the upward push (buoyant force) on the object matches the weight of the fluid displaced. This principle is the reason why ships float on water and why helium balloons rise in the air.

When considering skiers in avalanche conditions, this concept explains how they experience an upward force exerted by the snow, despite the density of snow being quite different from water.

Density

Density is a measure of how much mass is contained in a given volume. It is an important factor in understanding buoyancy and is defined by the formula: \( \text{Density} = \frac{\text{mass}}{\text{volume}} \).

For our skier problem, two densities are at play:

• The density of the snow, which is \( 96 \mathrm{~kg/m^3} \).
• The combined density of the skier, clothing, and equipment, which is \( 1020 \mathrm{~kg/m^3} \).

The significant difference in these values tells us how the skier's density compares to the snow's, influencing the magnitude of the buoyant force experienced.

A lower density implies that an object (like snow) has fewer or less tightly packed particles than something with a higher density (the skier in this case). This fundamental understanding is key when assessing if an object will float or sink in a fluid.

Gravitational Force

Gravitational force is the attractive force that pulls two masses toward one another. On Earth, this is experienced as the force that gives weight to physical objects.

The formula to calculate this force is \( F_g = \rho \times V \times g \), where

• \( \rho \): the density of the object (the skier in our discussion).
• \( V \): the volume of the object.
• \( g \): acceleration due to gravity, approximately \( 9.81 \mathrm{~m/s^2} \) on Earth.

In the scenario of a skier submerged in snow, the gravitational force calculates how strongly the skier is being pulled downward due to their mass and the force of gravity.

Understanding gravitational force is crucial in comparing with the buoyant force to determine if an object will float, sink, or remain neutrally buoyant.

Fluid Mechanics

Fluid mechanics is the branch of physics that studies the movement and behavior of fluids, which include liquids, gases, and plasmas. This area explores the forces and effects that accompany fluid motion.

In our skier example, fluid mechanics helps explain how snow behaves as a fluid to create a buoyant force. Key concepts from this field allow us to analyze how fluids flow and interact with submerged bodies.

Through fluid mechanics, we can study:

• Flow velocity of the snow as it moves in an avalanche.
• Pressure exerted by the fluid on objects and on the surroundings.
• Forces such as buoyancy that result from fluid interaction.

This knowledge is vital in applications such as predicting avalanche behavior, designing equipment to withstand such forces, and ensuring the safety of activities in snowy environments.