A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 one year ago
d^2r/dt^2 = (4pi^2r)/r^2
Find 2 scalar differential equations for x(t), y(t)
anonymous
 one year ago
d^2r/dt^2 = (4pi^2r)/r^2 Find 2 scalar differential equations for x(t), y(t)

This Question is Open

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\frac{ d^{2}\vec{r} }{ dt^{2} } = \frac{ 4\pi^{2}\hat{r} }{ \vec{r}^{2} }\]

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1i think you can get \(\ddot x(t) = 4 \pi^2 \frac{x}{\sqrt{x^2 + y^2}}\) just by ploughing through mechanically, and you get the same for y, pattern matched, and you'd then need polar to take it further. but i have no idea if that contextually makes any sense. looks like a repulsive radial field, but clutching at straws maybe.

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1@Michele_Laino will have an interesting opinion

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1by definition we have: \[\Large \hat r = \frac{{\vec r}}{r},\quad r = \left {\vec r} \right\] so, if we multiply both numerator and denominator of the right side, by r, we can write: \[\large \ddot \vec r = \frac{{{d^2}\vec r}}{{d{t^2}}} = 4{\pi ^2}\frac{{\vec r}}{{{r^3}}},\quad r = \sqrt {{x^2} + {y^2}} \]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1oops.. \[\Large \frac{{{d^2}\vec r}}{{d{t^2}}} = 4{\pi ^2}\frac{{\vec r}}{{{r^3}}},\quad r = \sqrt {{x^2} + {y^2}} \]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1now, using the cartesian coordinates, or the components of the vector r, we can write: \[\Large \vec r = \left( {x,y} \right)\] so substituting into the differential equation, we get: \[\Large \left( {\frac{{{d^2}x}}{{d{t^2}}},\frac{{{d^2}y}}{{d{t^2}}}} \right) = 4{\pi ^2}\frac{{\left( {x,y} \right)}}{{{{\left( {{x^2} + {y^2}} \right)}^{3/2}}}}\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1and finally we have the subsequent pair of differential equations: \[\Large \left\{ \begin{gathered} \frac{{{d^2}x}}{{d{t^2}}} = 4{\pi ^2}\frac{x}{{{{\left( {{x^2} + {y^2}} \right)}^{3/2}}}} \hfill \\ \hfill \\ \frac{{{d^2}y}}{{d{t^2}}} = 4{\pi ^2}\frac{y}{{{{\left( {{x^2} + {y^2}} \right)}^{3/2}}}} \hfill \\ \end{gathered} \right.\]
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.