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Rotational Motion (Neutron Star)

  1. Nov 14, 2016 #1
    1. The problem statement, all variables and given/known data
    I'm doing a question from a past paper, preparing for an upcoming exam. There is no solutions so I was wondering if my answer is correct for all parts.

    Take a star to be a uniform sphere with mass [tex]M_{i}=3.0 \times 10^{30} Kg[/tex] and radius [tex]R_{i} = 7.0 \times 10^{8}m[/tex] that rotates with a period of 27.0 days.

    (i) What's the star's angular speed of rotation?
    (ii) What's the star's angular momentum?
    (iii) At the end of it's lifetime, the star collapses to form a very compact star called a neutron star. If the star retains all it's mass when it collapses and the angular speed of the rotation of the neutron star that forms is [tex]\omega _{f} = 10^{4} rad/s[/tex]. what is the radius of the neutron star in kilometres?
    (iv) If the neutron star's period of rotation is observed to be increasing at a rate of [tex]1.2 \times 10^{-5}s/yr[/tex], what is the torque acting on the star?

    2. Relevant equations


    3. The attempt at a solution
    (i) [tex]27 days = 2332800 s[/tex]. The star rotates 360 degrees in 27 days. [tex] \omega = \frac{\frac{360}{2\pi}}{T} = 2.46 \times 10^{-5} rad/s [/tex]

    (ii) [tex] \begin{align} L & =I \omega \\
    & = \frac{2}{5} M_{i}R_{i}^{2} \omega \\
    & = \frac{2}{5}(3 \times 10^{30})(7 \times 10^{8})^{2} \\
    & = 1.446 \times 10^{43} Kgm^{2}/s\end{align} [/tex]

    (iii) Conservation of angular momentum [tex] \begin{align} I_{i} \omega_{i}&= I_{f} \omega_{f} \\
    \frac{2}{5}M_{i}R_{i}^{2} \omega_{i} & = \frac{2}{5}M_{f}R_{f}^{2} \omega_{f} \\
    R_{f} & = \sqrt{\frac{\omega_{i}}{\omega_{f}}R_{i}^{2}} \\
    & = 34.718.87 m \\
    & = 34.71 km \end{align} [/tex]

    (iv) [tex] \begin{align} \Delta L & = I_{f} \omega_{f} - I_{i} \omega{i} \\
    & = 1.2 \times 10^{34} kgm^{2}/s\end{align} [/tex]

    [tex] \tau = \frac{\Delta L}{\Delta T} = \frac{1.2 \times 10^{34}}{1.2 \times 10^{-5}} = 1 \times 10^{39} N [/tex]

    Thanks. Does this look okay?
     
  2. jcsd
  3. Nov 14, 2016 #2

    gneill

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    Staff: Mentor

    Revisit your determination of the angular velocity in part (i). The method should not involve degrees at all. One "cycle" or rotation is ##2 \pi ~ rad## (just as one rotation is also 360 degrees, but your angular velocity should be specified in radians per second).

    I believe that part (iv) is meant to pertain to the star once it's already in neutron star form: it's observed to be slowing at the specified rate. So it won't involve the initial state of the star at all.
     
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