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 $10 \mathrm{~m}$ and carries two people, $70 \mathrm{~kg}$ 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. FIGURE P1–101(Figure Cant Copy)

Key Concepts

Volume of a Sphere

The volume of a sphere is obtained using the formula V = (4/3)?r³, where r is the radius of the sphere. This mathematical concept is critical when determining how much fluid (such as air) a spherical object displaces, which in turn allows for the calculation of the buoyant force acting on that object.

Newton’s Second Law of Motion

Newton’s Second Law states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass (a = F_net/m). This law is used to find the resulting acceleration of a system when the upward buoyant force and the downward gravitational forces (from the mass of the object and any additional load) are known.

Archimedes’ Principle

This principle states that any object wholly or partially immersed in a fluid experiences an upward force equal to the weight of the fluid that is displaced by the object. In the context of buoyancy, it provides the basis for calculating the lift produced by a balloon as it displaces air, regardless of the specific gas used inside the balloon.

Buoyant Force

Buoyant force is the net upward force exerted by a fluid on a body immersed in it. It is calculated by multiplying the density of the fluid, the gravitational acceleration, and the volume of the displaced fluid. This concept is essential for understanding how objects can float or rise in a fluid, and it is directly applied in scenarios involving lighter-than-air gases like helium.

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