Balloons are often filled with helium gas because it weighs...

Balloons are often filled with helium gas because it weighs only about one-seventh of what air weighs under identical conditions. The buoyancy force, which can be expressed as $F_{b}=\rho_{\text {air }} g V_{\text {balloon }},$ will push the balloon upward. If the balloon has a diameter of $12 \mathrm{m}$ and carries two people, $85 \mathrm{kg}$ each, determine the acceleration of the balloon when it is first released. Assume the density of air is $\rho=1.16 \mathrm{kg} / \mathrm{m}^{3},$ and neglect the weight of the ropes and the cage.

Key Concepts

Newton's Second Law

Newton's Second Law states that the net force acting on an object equals the mass of the object multiplied by its acceleration. In the context of buoyancy problems, after computing the upward buoyant force and subtracting the weight of the object, Newton's Second Law is applied to determine the resulting acceleration.

Buoyancy Force

The buoyancy force is the upward force exerted by a fluid on an object immersed in it. This force is equal to the weight of the fluid displaced by the object, and it is determined by the fluid's density, the volume of the displaced fluid, and the acceleration due to gravity. This concept is essential in analyzing whether objects, such as balloons, will rise or sink in a fluid.

Archimedes' Principle

Archimedes' Principle is a fundamental concept in fluid mechanics stating that any object submerged in a fluid experiences an upward force equal to the weight of the fluid that the object displaces. This principle underpins the calculation of buoyant forces in various applications, from floating ships to rising balloons.

Volume Calculation for Spherical Objects

Calculating the volume of a sphere is often necessary in problems involving buoyancy when the object in question has a spherical shape. The volume, determined by the formula (4/3)?r³, is used to compute the amount of displaced fluid, which in turn is used to calculate the buoyant force on the object.

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