What must be the volume V of a 5.0 -kg balloon filled with...

What must be the volume $V$ of a $5.0$ -kg balloon filled with helium $\left(\rho_{\mathrm{He}}=0.178 \mathrm{~kg} / \mathrm{m}^{3}\right)$ if it is to lift a $30-\mathrm{kg}$ load? Use $\rho_{\text {air }}=1.29 \mathrm{~kg} / \mathrm{m}^{3}$. The buoyant force, $V \rho_{\mathrm{air}} g$, must lift the weight of the balloon, its load, and the helium within it: $$ V \rho_{\text {air }} g=(35 \mathrm{~kg})(g)+V \rho_{\mathrm{He}} g $$ which gives $$ V=\frac{35 \mathrm{~kg}}{\rho_{\mathrm{air}}-\rho_{\mathrm{He}}}=\frac{35 \mathrm{~kg}}{1.11 \mathrm{~kg} / \mathrm{m}^{3}}=32 \mathrm{~m}^{3} $$

What must be the volume $V$ of a $5.0$ -kg balloon filled with helium $\left(\rho_{\mathrm{He}}=0.178 \mathrm{~kg} / \mathrm{m}^{3}\right)$ if it is to lift a $30-\mathrm{kg}$ load? Use $\rho_{\text {air }}=1.29 \mathrm{~kg} / \mathrm{m}^{3}$.
The buoyant force, $V \rho_{\mathrm{air}} g$, must lift the weight of the balloon, its load, and the helium within it:
$$
V \rho_{\text {air }} g=(35 \mathrm{~kg})(g)+V \rho_{\mathrm{He}} g
$$
which gives
$$
V=\frac{35 \mathrm{~kg}}{\rho_{\mathrm{air}}-\rho_{\mathrm{He}}}=\frac{35 \mathrm{~kg}}{1.11 \mathrm{~kg} / \mathrm{m}^{3}}=32 \mathrm{~m}^{3}
$$

Key Concepts

Archimedes' Principle

Archimedes' Principle states that any object wholly or partially immersed in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid that is displaced by the object. This principle is essential for solving problems related to flotation and buoyancy.

Density

Density, defined as mass per unit volume, is a crucial concept in buoyancy problems because it determines both the weight of the displaced fluid and the weight of the object or gas. Understanding density differences allows one to determine whether an object will float or sink in a fluid.

Force Equilibrium

Force equilibrium is the condition where the sum of all forces acting on a system is zero, resulting in no net acceleration. In the context of buoyancy, the upward buoyant force must balance the downward forces (such as weight of the object and any additional loads) for the system to be in equilibrium.

Algebraic Problem Solving

Algebraic problem solving involves setting up an equation based on the physical principles at work (such as balancing forces) and then manipulating the equation to solve for the unknown variable, such as volume. This approach is widely applicable in various areas of physics and engineering problems.

Buoyant Force

The buoyant force is the upward force exerted by a fluid on any object immersed in it, which is equal to the weight of the fluid displaced by the object. This concept is fundamental in understanding how objects float or are lifted in fluids.

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