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Chapter 4: Problem 56

\(\bullet$$\bullet$$\bullet\) A hot-air balloon consists of a basket, one passenger, and some cargo. Let the total mass be \(M .\) Even though there is an upward lift force on the balloon, the balloon is initially accelerating downward at a rate of \(g / 3 .\) (a) Draw a free-body diagram for the descending balloon. (b) Find the upward lift force in terms of the initial total weight \(M g .\) (c) The passenger notices that he is heading straight for a waterfall and decides he needs to go up. What fraction of the total weight must he drop overboard so that the balloon accelerates upward at a rate of \(g / 2 ?\) Assume that the upward lift force remains the same.

Short Answer

Expert verified
Drop \( \frac{5}{9} \) of the initial total weight.

Step by step solution

01

Free-Body Diagram

To draw the free-body diagram for the balloon when it is accelerating downward, identify the forces acting. There are two main forces:1. The downward weight of the balloon, basket, passenger, and cargo, which is represented as weight force, \( W = M \cdot g \).2. The upward lift force, \( F_l \), acting on the balloon.Since it is accelerating downward, the net force is downward, which means that the weight force is greater than the lift force.
02

Finding the Upward Lift Force

To find the lift force, apply Newton's second law which states that \( F_{net} = M \cdot a \). Here, the acceleration \( a = \frac{g}{3} \) downward, so:\[F_{net} = M \cdot \left(-\frac{g}{3}\right) = F_l - M \cdot g\]Rearrange to solve for \( F_l \):\[F_l = M \cdot g - M \cdot \left(-\frac{g}{3}\right) = M \cdot g - \frac{M \cdot g}{3} = \frac{2M \cdot g}{3}\]Thus, the lift force \( F_l = \frac{2}{3} \cdot M \cdot g \).
03

Determine the Weight to Drop

To make the balloon ascend with an upward acceleration \( a = \frac{g}{2} \), set up the equation with the new conditions:Let the new mass be \( M' \). The net force is upward, so:\[F_l - M' \cdot g = M' \cdot \frac{g}{2}\]Using \( F_l = \frac{2}{3}M \cdot g \), substitute in:\[\frac{2}{3}M \cdot g - M' \cdot g = M' \cdot \frac{g}{2}\]Solve for \( M' \):\[\frac{2}{3}M \cdot g = \left( M' + \frac{M'}{2} \right) \cdot g \quad \Rightarrow \quad \frac{2}{3}M = \frac{3}{2}M'\]Rearrange and solve for \( M' \):\[M' = \frac{4}{9}M\]So, the fraction of weight to drop is:\[M - M' = M - \frac{4}{9}M = \frac{5}{9}M\]Thus, \( \frac{5}{9} \) of the initial total weight must be dropped.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Free-Body Diagram
A free-body diagram is a useful tool in physics for visualizing the forces acting on an object. To draw one for the hot-air balloon during its downward acceleration, you need to identify all acting forces. In this scenario, there are two main forces:
  • Weight Force (\( W = M \cdot g \)
  • Upward Lift Force (\( F_l \)
The weight force (\( W \)) pulls the balloon downwards, while the lift force (\( F_l \)) pushes it upwards. As the balloon accelerates downward at \( \frac{g}{3} \), the weight force is larger than the lift force. This indicates that the net force is downward. The diagram should clearly show an arrow pointing down for the weight force and a smaller arrow going up for the lift force. This graphical representation helps in understanding the underlying dynamics governing the balloon's motion.
Upward Lift Force
The upward lift force is the force that counteracts the downward pull of gravity on the balloon. To find its value, we apply Newton's Second Law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (\( F_{net} = M \cdot a \)). Here, the acceleration \( a \) is downward at \( \frac{g}{3} \).
According to this law:
  • The net force is downward, expressed as \( F_{net} = M \cdot \left(-\frac{g}{3}\right) \)
  • The equation becomes \( F_l - M \cdot g = M \cdot \left(-\frac{g}{3}\right) \)
  • Rearrange to solve for the upward lift force \( F_l = M \cdot g - \frac{M \cdot g}{3} = \frac{2}{3} \cdot M \cdot g \)
This means that even with acceleration downwards, the balloon experiences a significant lift, which is two-thirds of its total weight.
Net Force Calculation
Net force calculation is crucial when determining the changes needed for the balloon to ascend. To achieve an upward acceleration of \( \frac{g}{2} \), a new mass (\( M' \)) must be calculated. Assuming the lift force \( F_l \) remains stable, we need to:
  • Consider that \( F_l - M' \cdot g = M' \cdot \frac{g}{2} \)
  • Substitute \( F_l = \frac{2}{3} \cdot M \cdot g \)
  • The equation simplifies to \( \frac{2}{3}M \cdot g - M' \cdot g = M' \cdot \frac{g}{2} \)
  • Solving provides \( M' = \frac{4}{9}M \)
The passenger must drop \( \frac{5}{9} \) of the weight to adjust the mass from \( M \) to \( M' \), enabling ascent. This fraction ensures the net force is enough to overcome gravity, allowing acceleration upwards. Understanding this calculation helps you grasp how forces and mass interact to change the motion direction of the balloon.

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Most popular questions from this chapter

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