Insights How to Apply Newton’s Second Law to Variable Mass Systems

AI Thread Summary
Newton's second law, expressed as dP/dt = F_ext, can lead to confusion when applied to variable mass systems like rockets. In the example of a hovering rocket expelling combustion gases, a novice may incorrectly apply the equation, resulting in an impasse. The key condition for the rocket to maintain its position involves balancing the forces, specifically the thrust generated by the expelled gases against gravitational force. Understanding the nuances of variable mass systems is crucial for correctly applying Newton's second law. Proper application ensures accurate predictions of system behavior in dynamic scenarios.
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Introduction
The applicability of Newton’s second law in the oft-quoted “general form”  $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$ was an issue in a recent thread (see post #4) in cases of systems with variable mass.  The following example illustrates the kind of confusion that could arise from the (mis)application of Equation (1):
A rocket is hovering in place above ground near the Earth’s surface. Assume that the combustion gases are expelled at constant rate ##\beta=dm/dt## with velocity ##w## relative to the rocket.  What condition must hold for the rocket to hover in place?
A novice might start with Equation (1) and go down the garden path only to reach a quick impasse as shown below.
Attempted solution
We start with the general form of Newton’s second law, Equation (1) $$\frac{dP}{dt}=M\frac{dV}{dt}+V\frac{dM}{dt}=-Mg$$ If...

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