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Physics
Classical Physics
Optics
Central force and acceleration in the polar direction
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[QUOTE="Nugatory, post: 5441549, member: 382138"] Before you take on elliptical orbits in a central force, consider the simpler case of no force at all. What is the equation of motion of a particle that starts at the point ##(\theta=0,r=1))## with a speed ##v## in the tangential direction? It's not subject to any force or acceleration at all, yet neither the ##\theta## nor the ##r## components of its velocity are constant. Cartesian coordinates have the nice property that the basis vectors are functions of neither position nor time, so when you rewrite ##\vec{F}=m\vec{a}## as the differential equation ##\frac{d^2\vec{r}(t)}{dt^2}=\frac{\vec{F}}{m}## it simplifies into ##\frac{d^2A_x(t)}{dt^2}=\frac{F_x}{m}## (and likewise for the y and z components) and you can safely conclude that if ##F_x=0## then ##v_x## will be a constant. More general coordinate systems don't necessarily work that way. [/QUOTE]
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Forums
Physics
Classical Physics
Optics
Central force and acceleration in the polar direction
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