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The equation of motion F = ma in polar coordinates (r, o) takes the form F₁ = m(r-ro²), m(ř m(rø + 2ro), Fo = where F, and F are the radial and tangential (angular) components of the force F. Consider a particle that feels an angular force only, of the form F = mro. (There is nothing physical about this force; it simply makes the equations solvable.) Show that = √Alogr+B, where A and B are constants of integration, determined by the initial conditions. (Note that you will also need the equation F, = 0 to solve this system.) Hint: There are two differential equations. You can simplify the problem by defining w = and v, = r. Integrate one of the equations to find w as a function of r. You can then eliminate w from the second differential equation, and integrate with respect to r to find u, as a function of r, which is the form of the given answer.

The equation of motion F = ma in polar coordinates (r, o) takes the form F₁ = m(r-ro²), m(ř m(rø + 2ro), Fo = where F, and F are the radial and tangential (angular) components of the force F. Consider a particle that feels an angular force only, of the form F = mro. (There is nothing physical about this force; it simply makes the equations solvable.) Show that = √Alogr+B, where A and B are constants of integration, determined by the initial conditions. (Note that you will also need the equation F, = 0 to solve this system.) Hint: There are two differential equations. You can simplify the problem by defining w = and v, = r. Integrate one of the equations to find w as a function of r. You can then eliminate w from the second differential equation, and integrate with respect to r to find u, as a function of r, which is the form of the given answer.

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1 `The equation of motion F = ma in polar coordinates (r, o) takes the form m(* - ro²), m(rø + 2ro), Fr = Fø = where F, and Fo are the radial and tangential (angular) components of the force F. Consider a particle that feels an angular force only, of the form F = mro. (There is nothing physical about this force; it simply makes the equations solvable.) Show that r = √Alogr+ B, where A and B are constants of integration, determined by the initial conditions. (Note that you will also need the equation F, = 0 to solve this system.) Hint: There are two differential equations. You can simplify the problem by defining w = 6 and v, = r. Integrate one of the equations to find w as a function of r. You can then eliminate w from the second differential equation, and integrate with respect to r to find u, as a function of r, which is the form of the given answer.
Transcribed Image Text:1 `The equation of motion F = ma in polar coordinates (r, o) takes the form m(* - ro²), m(rø + 2ro), Fr = Fø = where F, and Fo are the radial and tangential (angular) components of the force F. Consider a particle that feels an angular force only, of the form F = mro. (There is nothing physical about this force; it simply makes the equations solvable.) Show that r = √Alogr+ B, where A and B are constants of integration, determined by the initial conditions. (Note that you will also need the equation F, = 0 to solve this system.) Hint: There are two differential equations. You can simplify the problem by defining w = 6 and v, = r. Integrate one of the equations to find w as a function of r. You can then eliminate w from the second differential equation, and integrate with respect to r to find u, as a function of r, which is the form of the given answer.
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