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Question

The Sun's mass is

2.0×1030kg,2.0 × 10^{30} kg,

its radius is

7.0×105km,7.0 × 10^5 km,

and it has a rotational period of approximately 28 days. If the Sun should collapse into a white dwarf of radius 3.5 × 10³ km, what would its period be if no mass were ejected and a sphere of uniform density can model the Sun both before and after?

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From conservation of angular momentum \textbf{conservation of angular momentum } we know that :  the angular momentum of a system of particles around a point in a fixed inertial reference frame is conserved if there is no net torque around this point \textbf{ the angular momentum of a system of particles around a point in a fixed inertial reference frame is conserved if there is no net torque around this point }.

τ=dLdt=0\sum \tau = \dfrac{d L }{dt} =0

Li=LfIiωi=Ifωf\begin{gather*} L_{i} = L_{f} \\ I_{i} \omega_{i} = I_{f} \omega_{f} \\ \end{gather*}

From the moment of inertia of the sun before it collapses \textbf{the moment of inertia of the sun before it collapses } we know that :

Ii=mri2I_{i} = m r_{i}^2

The moment of inertia of the sun after it collapses \textbf{The moment of inertia of the sun after it collapses } we know that :

If=mrf2I_{f} = m r_{f}^2

And we know that :

T=2πωωi=2 πTi=2 π28T = \dfrac{2 \pi}{\omega} \quad \rightarrow \quad \omega_{i} = \dfrac{ 2\ \pi }{T_{i}} = {2\ \pi }{28 }

So ,

Iiωi=IfωfIi2 πTi=If2 πTfTf=2 πIfTi2 πIiTf=TiIfIi\begin{gather*} I_{i} \omega_{i} = I_{f} \omega_{f} \\ I_{i} \dfrac{2\ \pi }{T_{i} } = I_{f} \dfrac{2\ \pi }{T_{f} } \\ T_{f} = 2 \ \pi I_{f} \cdot \dfrac{ T_{i} }{2\ \pi I_{i}}\\ T_{f} = \dfrac{ T_{i} I_{f}}{I_{i}} \end{gather*}

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