Problem 56 - University Physics with Modern Physics

Question

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 Mg. (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.

Step-by-Step Solution

Answer

(a) The forces are weight downward and lift upward. (b) Lift force is \( \frac{2}{3} Mg \). (c) Drop \( \frac{5}{9} \) of the weight.

Step 1: Understanding the Free-Body Diagram

The free-body diagram for the descending balloon includes two forces: the gravitational force acting downward, which is the weight of the balloon, represented as \( Mg \), and the lift force acting upward represented as \( F_l \). These are the only forces acting along the vertical direction.

Step 2: Calculating the Lift Force

Using Newton's second law, \( F_{net} = ma \). Since the balloon accelerates downward at \( g/3 \), the net force \( F_{net} = M(-g/3) \). The weight acting downward is \( Mg \), so set up the equation: \( Mg - F_{l} = M \left( \frac{-g}{3} \right) \). Simplifying, we find \( F_l = Mg - \frac{Mg}{3} = \frac{2Mg}{3} \). Therefore, the lift force \( F_l = \frac{2}{3} Mg \).

Step 3: Determining the Required Weight Loss

To accelerate upwards at \( g/2 \), let the new total mass be \( M' \). The net force for upward acceleration is given by \( F_{net} = M' \frac{g}{2} \). Since we want to keep the lift force the same, \( F_l = \frac{2}{3}Mg \). Thus, the equation becomes \( \frac{2}{3}Mg - M'g = M' \frac{g}{2} \). Solving for \( M' \), we get: \( \frac{2}{3}Mg = \frac{3}{2}M'g \). Hence, \( M' = \frac{4}{9}M \). The fraction of the total weight that needs to be dropped is \( 1 - \frac{4}{9} = \frac{5}{9} \).

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that connects force, mass, and acceleration. It is often written as \( F = ma \), where \( F \) is the net force applied to an object, \( m \) is the mass of the object, and \( a \) is the acceleration produced. This formula tells us that the acceleration of an object depends directly on the net force acting on it and inversely on its mass.
This law is crucial in understanding motion because it allows us to predict how an object will move when forces are applied. In the context of the hot-air balloon problem, Newton's Second Law helps us figure out the net force involved, which includes both the lift force and the gravitational force. By calculating these forces, we can determine the resulting acceleration of the balloon. This understanding is foundational in physics problem solving as it breaks down interactions into measurable components.

Free-Body Diagrams

Free-Body Diagrams (FBDs) are visual tools used in physics to represent all the forces acting on a single object. This simplification enables problem solvers to analyze an object's motion without distractions from outside surroundings.
In the hot-air balloon scenario, the FBD would depict two main forces: the gravitational force directed downward, portrayed by \( Mg \), and the upward lift force, depicted by \( F_l \).
The FBD helps us visualize how these forces compete, leading to the balloon's downward acceleration. By clearly understanding these forces and their directions, we can better apply Newton's Second Law to solve for unknowns like the lift force. Creating accurate FBDs is an essential skill for physics students, as they bridge the gap between conceptual understanding and mathematical analysis.

Upward Lift Force

The upward lift force in this problem is vital for counterbalancing the downward gravitational pull on the hot-air balloon. Lift force \( F_l \) plays a central role because it determines whether the balloon rises or descends. Initially, the balloon descends because the gravitational force exceeds the lift force.
According to the step by step solution, the lift force has been calculated using Newton's Second Law. Given the downward acceleration of \( \frac{g}{3} \), we know that the net force results from the gravitational force overpowering the lift. By setting up the equation \( Mg - F_l = M \left( \frac{-g}{3} \right) \), we learn that the lift force amounts to \( \frac{2}{3}Mg \).
Understanding the lift force is crucial because it helps us comprehend how much force is needed not just to counteract gravity, but also to achieve specific accelerations for ascending or descending.

Acceleration Calculations

Calculating acceleration is essential for determining how the velocity of an object changes over time under the influence of applied forces. In physics, acceleration can be derived from the net force acting over an object's mass, using the formula \( a = \frac{F}{m} \).
In the hot-air balloon example, we are tasked with changing the balloon's downward motion to upward motion by adjusting its acceleration. To achieve this, we must understand how changing the mass affects the required force to cause a particular acceleration. The original acceleration downward is \( \frac{g}{3} \), but to move upward at \( \frac{g}{2} \), the balloon must drop some load.
The acceleration change illustrates how altering the system's mass changes the dynamics. By correctly calculating the exact mass to be offloaded, using \( M' = \frac{4}{9}M \), allows for overcoming the gravitational pull. Acceleration calculations are pivotal for planning and manipulating motion in a structured manner.