Saturday, May 7, 2016

homework and exercises - Variable mass dynamics: Particle and Rigid Body


I'm encountering some issues in the understanding of some basic concepts about the dynamics of variable-mass particles and rigid bodies.


For what I found, for example reading On the use and abuse of Newton's Second Law for Variable Mass Problems (Plastino,Muzzio) and also Lectures On Theoretical Physics: Mechanics (Sommerfield -- p28) the second law of Dynamics is not suitable for a variable mass particle; instead you should use the momentum conservation:




e.g. Rocket:


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Applying the conservation of momentum for an isolated system:


$$p_t=mv$$ $$p_{t+\Delta t}=(m-\text{d}m)(v+\text{d}v) +\text{d}m\,(v-u_e)$$


$$ \frac{\text{d}p}{\text{d}t}=\frac{mv + mdv -vdm +v\text{d}m -u_edm - mv}{\text{d}t}=0$$ $$ 0 =\frac{mdv-u_edm}{\text{d}t}$$ $$ dv=-u_e\frac{dm}{m}$$


that is the classical Tsiolkovsky rocket equation, that can be integrated $\int_{t_0}^t$: $$\boxed{\Delta v = u_e\ln\frac{m_0}{m}}$$


Where $u_e$ is the velocity of the gases that are coming out from the nozzle.


In other books the same equation is obtained from Newton's Law: $$ m\frac{\text{d}v}{\text{d}t} = \sum_i F^{\text{ext}}_i$$ where the external forces are basically just the thrust (simplest case) that is given by: $$ T = \dot{m}u_e + A_e(p_e - p_a) = \dot{m}c$$


where $A_e$ is the area of outflow for the nozzle, $p_e$ the outflow pressure, $p_a$ the ambient pressure (hence $c=u_e + \frac{A_e(p_e - p_a)}{\dot{m}}$ is the equivalent velocity)


substituting: $$m\frac{\text{d}v}{\text{d}t} = \dot{m}c$$ being $\dot{m} = -\frac{\text{d}m}{\text{d}t}$ because of the mass loss we get: $$ \text{d}v = -c\frac{\text{d}m}{m}$$


that is basically the same equation but with the equivalent velocity instead of the real convective gas velocity. This is the first confusing passage...





And what about rigid bodies?


The equation would be: $$ \frac{\text{d}\mathbf{Q}}{\text{d}t} =\sum_i \mathbf{F_i^\text{ext}}$$ so should I do: $$ \frac{\text{d}\mathbf{Q}}{\text{d}t} = m\mathbf{\dot{v}} + \mathbf{v}\dot{m}=\sum_i \mathbf{F_i^\text{ext}}$$


or not? I'm really confused about that.




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