Tidal Forces near a Black Hole.An astronaut inside a spacecraft,

which protects her from harmful radiation, is orbiting a black hole at a distance

of 120 km from its centre. The black hole is 5.00 times the mass of the sun and

has aSchwarzschild radius of 15.0 km. The astronaut is positioned inside the

spaceship such that one of her 0.030-kg ears is 6.0 cm farther from the black

hole than the centre of mass of the spacecraft and the other ear is 6.0 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 centre of

gravity of her head at the same point as the centre of

mass? Explain.

Step by step solution

01

Identification of given data

• The orbit radius is,r = 120km
• The distance of orbit radius from either ear is,$\delta$= 6cm
• The mass of each ear,m.= 0.030kg

02

Concept of gravitational force

The gravitational pull exerted by the Earth's mass on all bodies in the vicinity of its surface.

The gravitational force can be given by,

$F=\frac{GMm}{r^2}$… (i)

03

Determine tension between two ears

The Newton’s second law of motion is,

$F=ma$… (ii)

Equating equation (i) and equation (ii),

$\frac{GMm}{\left(r+\delta \right)}-F=m{\omega }^2\left(r+s\right)$

Here, $\delta$can be either of the sign.

Replace $m{\omega }^2$for $\delta =0$,

$m{\omega }^2=\frac{GMm}{r^3}$

The tension between ears can be evaluated by,

$$\begin{gathered} F=\left(GMm\right)\left[\frac{r+\delta }{r^3}-\frac{1}{{\left(r+\delta \right)}^2}\right] \\ F=\frac{GMm}{r^2}\left[r+\delta +r\left(1+{\left(\frac{\delta }{r}\right)}^{-2}\right)\right] \end{gathered}$$

Expanding above equation using Binomial theorem,

$$\begin{gathered} F=\frac{GMm}{r^3}\left[r+\delta +r\left(1-2\left(\frac{\delta }{r}\right)\right)\right] \\ F=\frac{GMm}{r^3}\left(3\delta \right) \\ F=2.1N \end{gathered}$$

Thus, the tension between two ears is,2.1N.

04

Evaluate the centre of gravity on ears

The centre of gravity does not correspond to the centre of mass, as previously

stated. The gravitational pull on each ear is different.

Thus, the gravity force of two ears will not be same.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!