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Math Help - central force problem

  1. #1
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    central force problem

    A particle of unit mass moves in a plane under a central force lamda^2 / r^5 where lamda is a constant.

    Explain why (r dot)^2 + r^2.(theta dot)^2 - lamda^2 / (2r^4) = constant
    done this part

    The particle is projected from a point P at which r = a with speed lamda / (root2 a^2). Find a differential equation for r as a function of the polar angle theta along the orbit, and show that the orbit is a circle through O.

    I'm not entirely sure how to start this part

    my initial thought was to put the IC's into the above equation. But this simply gives me a constant of zero and then I cant see how to progress from there..

    many thanks
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  2. #2
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    Quote Originally Posted by James0502 View Post
    A particle of unit mass moves in a plane under a central force lamda^2 / r^5 where lamda is a constant.

    Explain why (r dot)^2 + r^2.(theta dot)^2 - lamda^2 / (2r^4) = constant
    done this part

    The particle is projected from a point P at which r = a with speed lamda / (root2 a^2). Find a differential equation for r as a function of the polar angle theta along the orbit, and show that the orbit is a circle through O.

    I'm not entirely sure how to start this part

    my initial thought was to put the IC's into the above equation. But this simply gives me a constant of zero and then I cant see how to progress from there..

    many thanks
    Yes the constant is zero. So you have:

    (\dot{r})^2 + r^2 (\dot{\theta})^2 - \frac{\lambda^2}{2 r^4} = 0 .... (1)

    Now you should also know that

    r^2 \dot{\theta} = C .... (2)

    where C is a constant (conservation of angular momentum).

    I think you might need to know the angle that the initial velocity makes with \theta = 0 before you can calculate the value of C.

    Now note:

    \dot{r} = \frac{d r}{d \theta} \cdot \frac{d \theta}{d t}

    Substitute from equation (2):

    = \frac{d r}{d \theta} \cdot \frac{C}{r^2} = -C \frac{d}{d \theta} \left( \frac{1}{r} \right)

    Substitute u = \frac{1}{r}:

    = -C \frac{du}{d \theta} .... (3)


    Substitute u = \frac{1}{r}, equation (2) and equation (3) into equation (1) and simplify to get the required DE (well, it's a DE for u as a function of \theta - good enough).

    Now show that the solution to this DE is a circle (note: the solution is in polar form).
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