An astronaut inside a spacecraft, which protects her from harmful radiation, is orbiting a black hole at

a distance of $13,500$ km from its event horizon. The black hole is $5250$ times the mass of the sun. $($a$)$

Show that it has a Schwarzchild radius of approximately $1\mathrm{.}55104$ km$.$ The astronaut is positioned

inside the spaceship such that one of her $0\mathrm{.}0350$ kg ears is $6\mathrm{.}3$ cm farther from the black hole than the

center of mass of the spacecraft and the other ear is $6\mathrm{.}3$ cm closer. $($a$)$ What is the tension between her

ears? Would the astronaut find it difficult to keep from being torn apart by the gravitational forces?

$($Since her whole body orbits with the same angular velocity, one ear is moving too slowly for the radius

of its orbit and the other is moving too fast. Hence her head must exert forces on her ears to keep them

in their orbits.$)($b$)$ Is the center of gravity of her head at the same point as the center of mass? Explain.