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I am supposed to integrate Kepler´s second Law to find T (period) as a function of R (radius of a circle). However, this is a special case where we talk about a circle and not about an ellipse. See the attachment for details.
 one year ago
 one year ago
I am supposed to integrate Kepler´s second Law to find T (period) as a function of R (radius of a circle). However, this is a special case where we talk about a circle and not about an ellipse. See the attachment for details.
 one year ago
 one year ago

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TomLikesPhysicsBest ResponseYou've already chosen the best response.0
Here is what I did so far.
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
But I have to no good reason to drop the sin when I integrate. However I think the end is correct. But I can not justify my steps to get there.
 one year ago

ivanmlernerBest ResponseYou've already chosen the best response.0
What did you get when you integrated?
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
The thing you see on the sol.jepg picture. On the left side it will be A(T) which is a full circle because we are talking about a circular orbit so when the Period T passes the planet is where it was before and has therefore made a full circle area which is pi*r^2. On the right side I am kind of clueless how to integrate that stuff properly.
 one year ago

XishemBest ResponseYou've already chosen the best response.1
Alright, if we use the equation you're using: \[dA=\frac{1}{2}r v \sin(\alpha)dt\]Which of those quantities changes in time?
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
A and the angle alpha. r is constant because we talk about a circular orbit, and therefore the velocity is also constant (or the speed).
 one year ago

XishemBest ResponseYou've already chosen the best response.1
Now, I want you to be cautious. In this example, you know that the angle changes linearly in time, but you know exactly how it changes?
 one year ago

XishemBest ResponseYou've already chosen the best response.1
I'm curious how you derived your differential equation. Is that the equation that is given to you for Kepler's second law?
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
Yes, if you look at Kepler´s law and say that dt is very small than you can say that the area made by the radius is a triangle.
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
here is a picture of it
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
However, I am still clueless how the angle depends on time. Is it the integral of omega (the angular velocity)?
 one year ago

XishemBest ResponseYou've already chosen the best response.1
I'm not sure. Alpha is going to be a function of r(t) and r(t+dt).
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
That one should be helpfull...
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
than I could rewrite 0.5*v*r*sin(alpha) as 0.5*omega*sin(omega*t) but the integral of that after dt is only 0.5*cos(omega*t)
 one year ago

XishemBest ResponseYou've already chosen the best response.1
Alpha, as an angle, should be the radial angle if it's omega*t. dw:1352577895185:dw
 one year ago

XishemBest ResponseYou've already chosen the best response.1
Because the angular velocity is defined as: \[\omega =\frac{d \theta}{dt} \rightarrow \omega\ dt = d \theta\]
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
but that gives me:
 one year ago

XishemBest ResponseYou've already chosen the best response.1
I would do it this way. Since this is a circle, you can define the area of any arc as: \[A=\frac{1}{2}\theta r^2\]If we look at a small piece of A, then it becomes:\[dA=\frac{1}{2}r^2d\theta\]\[dA=\frac{1}{2}r^2\omega dt\]
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
but I want to find T as a function of r. How does this help?
 one year ago

XishemBest ResponseYou've already chosen the best response.1
\[\int\limits_{A(0)}^{A(T)}dA=\int\limits_{0}^{T}\frac{1}{2}r\omega\ dt\]\[A(T)=\frac{1}{2}r\omega T\]\[\pi r^2=\frac{1}{2}r\omega T\]
 one year ago

XishemBest ResponseYou've already chosen the best response.1
Sorry, sorry. I missed an r term. One second.
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
It seems that we only used mathematics but not Kepler´s second law.
 one year ago

XishemBest ResponseYou've already chosen the best response.1
Well, Kepler's second law just states that the area swept out by the radius vector sweeps out equal areas in equal time intervals.
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
... ah so we used that to claim that omega is constant!
 one year ago

XishemBest ResponseYou've already chosen the best response.1
Usually, the more general form of Kepler's law that I use most is: \[dA=\frac{1}{2}\vec r \times d\vec r\]No, not exactly. We could only claim that because it's a circle.
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
But than we did not use any stuff of kepler. The formula for A follows from the unit circle and the rest was mathematics.
 one year ago

XishemBest ResponseYou've already chosen the best response.1
It's not that. It's that we made a simplification based on our understanding of the problem.
 one year ago

TomLikesPhysicsBest ResponseYou've already chosen the best response.0
Where did we simplify?
 one year ago
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