1. Tardigrade
  2. Question
  3. Physics
  4. Consider a nebula in the form of a ring of radius R and mass M. A star of mass m(m<< M) is located at distance R from the centre of the ring on its axis, initially at rest. The speed with which it crosses the centre of the ring is v=√(p-√q) (G M/R) . Find (p+q) .

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Q. Consider a nebula in the form of a ring of radius $R$ and mass $M$. A star of mass $m(m<< M)$ is located at distance $R$ from the centre of the ring on its axis, initially at rest. The speed with which it crosses the centre of the ring is $v=\sqrt{(p-\sqrt{q}) \frac{G M}{R}} .$ Find $(p+q) .$

Gravitation

Solution:

In the figure,
Total energy at $P=$ Total energy at $O$
image
$-\frac{G m M}{\sqrt{2} R}=\frac{1}{2} m v^{2}-\frac{G m M}{R}$
$\therefore v=\sqrt{(2-\sqrt{2}) \frac{G M}{R}}$