Questions: The balloon (including the helium within it) and the instruments it carries is 25.0 kg. The density of air at ground level is 1.29 kg / m^3. (a) What is the magnitude of the buoyant force (in N) acting on the balloon, just after it is released from ground level? 629.8 What is the definition of the buoyant force? How does it depend on volume? What fluid is displaced by the balloon?

Solution Steps

The buoyant force is the upward force exerted by a fluid on an object submerged in it. According to Archimedes' principle, the magnitude of the buoyant force is equal to the weight of the fluid displaced by the object. The formula for the buoyant force \( F_b \) is given by:

\[ F_b = \rho_{\text{fluid}} \cdot V \cdot g \]

where:

• \( \rho_{\text{fluid}} \) is the density of the fluid (in this case, air) in \(\text{kg/m}^3\),
• \( V \) is the volume of the fluid displaced by the object (in \(\text{m}^3\)),
• \( g \) is the acceleration due to gravity, approximately \(9.81 \, \text{m/s}^2\).

The fluid displaced by the balloon is air, as the balloon is released into the atmosphere. The density of air at ground level is given as \(1.29 \, \text{kg/m}^3\).

To find the buoyant force, we need to calculate the volume of air displaced by the balloon. However, the problem does not provide the volume directly. Instead, we can use the given buoyant force value to verify the calculation.

Given:

• The buoyant force \( F_b = 629.8 \, \text{N} \).

Using the formula for buoyant force:

\[ F_b = \rho_{\text{air}} \cdot V \cdot g \]

Rearranging to solve for \( V \):

\[ V = \frac{F_b}{\rho_{\text{air}} \cdot g} \]

Substituting the known values:

\[ V = \frac{629.8}{1.29 \cdot 9.81} \]

Calculating \( V \):

\[ V \approx \frac{629.8}{12.6549} \approx 49.77 \, \text{m}^3 \]

This volume is consistent with the buoyant force given.

Final Answer