A hot-air balloon is accelerating upward under the influence...

A hot-air balloon is accelerating upward under the influence of two forces, its weight and the buoyant force. For simplicity, consider the weight to be only that of the hot air within the balloon, thus ignoring the balloon fabric and the basket. (a) How is the weight of the hot air determined from a knowledge of its density Phot air and the volume $V$ of the balloon? (b) For a given volume $V$ of the balloon, does the buoyant force depend on the density of the hot air inside the balloon, the density of the cool air outside the balloon, or both? Provide a reason for your answer. (c) Draw a free-body diagram for the balloon, showing the forces that act on it. How is the upward acceleration of the balloon related to these forces and to its mass? The hot air inside the balloon has a density of $\rho_{\text {hot air }}=0.93 \mathrm{~kg} / \mathrm{m}^{3},$ and that of the cool air outside is $\rho_{\text {cool air }}=1.29 \mathrm{~kg} / \mathrm{m}^{3} .$ What is the acceleration of the rising balloon?

A hot-air balloon is accelerating upward under the influence of two forces, its weight and the buoyant force. For simplicity, consider the weight to be only that of the hot air within the balloon, thus ignoring the balloon fabric and the basket. (a) How is the weight of the hot air determined from a knowledge of its density Phot air and the volume $V$ of the balloon? (b) For a given volume $V$ of the balloon, does the buoyant force depend on the density of the hot air inside the balloon, the density of the cool air outside the balloon, or both? Provide a reason for your answer. (c) Draw a free-body diagram for the balloon, showing the forces that act on it. How is the upward acceleration of the balloon related to these forces and to its mass?
The hot air inside the balloon has a density of $\rho_{\text {hot air }}=0.93 \mathrm{~kg} / \mathrm{m}^{3},$ and that of the cool air outside is $\rho_{\text {cool air }}=1.29 \mathrm{~kg} / \mathrm{m}^{3} .$ What is the acceleration of the rising balloon?

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation used to show all the forces acting on an object. For the balloon, key forces include the downward weight of the hot air and the upward buoyant force from the displaced cool air. This diagram helps in visualizing the net force that causes the balloon to accelerate upward.

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 on it and inversely proportional to its mass. This principle is used to relate the difference between the buoyant force and the weight (net force) to the acceleration of the balloon, given the mass of the hot air inside.

Buoyant Force

Buoyant force is the upward force exerted by a fluid that opposes the weight of an immersed object, as described by Archimedes' principle. It depends on the density of the surrounding fluid (in this case, the cool air outside) and the volume of the displaced fluid, not on the contents (hot air) inside the balloon.

Density

Density is a measure of mass per unit volume. In this context, it is used to relate the amount of air (mass) contained within a specific volume of the hot-air balloon and the surrounding cool air, which in turn affects both the weight of the hot air inside and the buoyant force acting on the balloon.

Weight

Weight is the force exerted by gravity on a mass. In the scenario, the weight of the hot air within the balloon is determined by multiplying its density by the balloon's volume to get the mass, and then multiplying that mass by the acceleration due to gravity.

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