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

9.41\% of the gravitational force is offset by the buoyant force.

Step 1: Identify Forces at Play

The problem involves two primary forces: the gravitational force acting on the skier and the buoyant force exerted by the snow. Gravitational force \( F_g = \rho_s \cdot V \cdot g \), where \( \rho_s \) is the density of the skier, \( V \) is the volume, and \( g \) is the gravitational acceleration.

Step 2: Calculate Gravitational Force

The gravitational force on the skier is calculated using the skier's density \( \rho_s = 1020 \ \text{kg/m}^3 \), volume \( V \), and gravitational acceleration \( g = 9.8 \ \text{m/s}^2 \):\[ F_g = 1020 \, V \, g \]

Step 3: Calculate Buoyant Force

The buoyant force is calculated using the density of the snow \( \rho_{snow} = 96 \ \text{kg/m}^3 \), the same skier volume \( V \), and gravitational acceleration \( g \):\[ F_b = 96 \, V \, g \]

Step 4: Determine the Percentage Offset

The percentage of the gravitational force offset by the buoyant force is given by the ratio \( \left( \frac{F_b}{F_g} \right) \times 100\% \):\[ \text{Percentage} = \left( \frac{96 \, V \, g}{1020 \, V \, g} \right) \times 100\% = \left( \frac{96}{1020} \right) \times 100\% \]

Step 5: Calculate the Numerical Value

Compute the percentage:\[ \frac{96}{1020} = 0.0941 \]Therefore, the percentage of the gravitational force offset by the buoyant force is:\[ 0.0941 \times 100\% = 9.41\% \]

Key Concepts

Gravitational Force

Gravitational force is the force that attracts objects toward the center of the Earth, giving them weight. Imagine it like a gentle, invisible rope pulling everything toward the ground. In physics, it is calculated using the formula:

• \( F_g = \rho \cdot V \cdot g \)
• where \( \rho \) is the density of the object,
• \( V \) is the volume of the object,
• and \( g \) is the gravitational acceleration, approximately \( 9.8 \ \text{m/s}^2 \).

This force is what keeps us grounded and ensures that when you drop a ball, it falls straight to the floor. In the context of our skier, the gravitational force represents how much the skier, along with their gear, "weighs" due to the pull of gravity. It is calculated using the skier's density and volume along with the constant gravitational pull of Earth. Understanding this force is crucial when exploring other concepts like buoyancy.

Density

Density is a measure of how much mass is contained in a given volume. Think of it as how tightly packed the matter in an object is. It's why a stone sinks in water but a piece of wood floats. Density is calculated as:

• \( \rho = \frac{m}{V} \)
• where \( m \) is mass,
• \( V \) is volume.

The greater the density, the more mass is crammed into a specific volume. In our skier scenario, we have two densities to consider:

• The skier's density, which includes their body, clothing, and equipment, is \( 1020 \ \text{kg/m}^3 \).
• The snow density, which is lighter at \( 96 \ \text{kg/m}^3 \).

Comparing these densities helps us understand the interaction between the skier and the snow, especially in determining the buoyant force acting on the skier.

Buoyant Force

Buoyant force is a fascinating principle! It is the upward force that allows objects to float or at least feel lighter in a fluid. This force is created by the displacement of fluid when an object is submerged. Archimedes first described buoyancy, explaining why things float, sink, or remain suspended in fluids. The formula for finding the buoyant force is:

• \( F_b = \rho_{fluid} \cdot V \cdot g \)
• where \( \rho_{fluid} \) is the fluid density,
• \( V \) is the submerged volume of the object,
• and \( g \) is gravitational acceleration.

In our scenario, the snow acts as the fluid. The skier experiences an upward buoyant force from the snow, countering some of the gravitational pull trying to drag them down. This difference is crucial in understanding the balance of forces acting on the skier. In this specific case, the buoyant force offsets 9.41% of the skier’s gravitational force, which tells us snow has less density compared to the skier, offering support but not enough to counteract all gravitational pull. This concept is vital in various applications, from designing boats to understanding avalanches.