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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`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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b9c0895-132d-4cab-a151-ba0a5b8f5e11%2F0f0e3ca2-7fbb-4951-b04c-2653be8f7332%2Foulpmf8_processed.png&w=3840&q=75)
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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