The radius R h and mass M h of a black hole are related by R h = 2 G M h c 2 , where, c is the speed of light. Assume that the gravitational acceleration a g of an object at a distance r 0 = 1 . 001 R h from the centre of a black hole, is given by the equation, a g = G M r 2 (it is for large black holes). a In terms of M h , find a g at r 0 . b Does a g at r 0 increase or decrease as M h increases? c What is a g at r 0 for a very large black hole whose mass is 1 . 55 × 10 12 times the mass of Sun 1 . 99 × 10 30 kg ? d If an astronaut of height 1 . 66 m is at r 0 with her feet down, what is the difference in gravitational acceleration between her head and feet? e Is the tendency to stretch the astronaut severe?

a Plugging, R h = 2 G M h c 2 in the indicated expression, we find, a g = G M h 1 . 001 R h 2 = G M h ( 1 . 001 ) 2 2 G M h c 2 2 = c 4 ( 2 . 002 ) 2 G 1 M h This yields, a g = 3 . 02 × 10 43 kg m s - 2 M h b Since M h is in the denominator of the above result, a g decreases as M h increases. c With M h = 1 . 55 × 10 12 1 . 99 × 10 30 kg , we obtain, a g = 3 . 02 × 10 43 kg m s - 2 1 . 55 × 10 12 1 . 99 × 10 30 kg ⇒ a g = 9 .82 m s - 2 d This part refers specifically to the very large black hole treated in the previous part. With mass M and r = 2 . 002 G M c 2 , we obtain, d a g = - 2 G M 2 . 002 G M c - 2 3 d r = - 2 c 6 ( 2 . 002 ) 3 ( G M ) 2 d r where, d r → 1 . 66 m is the difference in acceleration at head and feet. This yields (in absolute value) an acceleration difference of 7 . 12 × 10 − 15 m s − 2 . e No,The miniscule result of the previous part implies that in this case, any effects due to the differences of the gravitational forces on the body are negligible.

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Principles Of Physics International Student Version>Chapter 13 - Gravitation>Problems>Q 38

An asteroid, whose mass is $2.0 \times 10^{- 4}$ times the mass of the Earth, revolves in a circular orbit around the Sun at a distance that is $3.0$ times Earth's distance from the Sun. $(a)$ Calculate the period of revolution of the asteroid in $a_{g} = \frac{G M}{r^{2}} = \frac{\frac{G M}{R_{E}^{2}}}{{(1 + \frac{h}{R_{E}})}^{2}} = \frac{g}{{(1 + \frac{h}{R_{E}})}^{2}}$ . $(b)$ What is the ratio of kinetic energy of the asteroid to the kinetic energy of Earth?

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Principles Of Physics International Student Version>Chapter 13 - Gravitation>Problems>Q 39

The radius

R_{h}

and mass

M_{h}

of a black hole are related by

R_{h} = \frac{2 G M_{h}}{c^{2}}

, where,

c

is the speed of light. Assume that the gravitational acceleration

a_{g}

of an object at a distance

r_{0} = 1.001 R_{h}

from the centre of a black hole, is given by the equation,

a_{g} = \frac{G M}{r^{2}}

(it is for large black holes).

(a)

In terms of

M_{h}

, find

a_{g}

r_{0}

(b)

Does

a_{g}

r_{0}

increase or decrease as

M_{h}

increases?

(c)

What is

a_{g}

r_{0}

for a very large black hole whose mass is

1.55 \times 10^{12}

times the mass of Sun

1.99 \times 10^{30} kg

(d)

If an astronaut of height

1.66 m

is at

r_{0}

with her feet down, what is the difference in gravitational acceleration between her head and feet?

(e)

Is the tendency to stretch the astronaut severe?

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Principles Of Physics International Student Version>Chapter 13 - Gravitation>Problems>Q 40

One way to attack a satellite in Earth's orbit is to launch a swarm of pellets in the same orbit as the satellite but in the opposite direction. Suppose a satellite in a circular orbit,

500 km

above the Earth's surface collides with a pellet having mass

5.0 g

(a)

What is the kinetic energy of the pellet in the reference frame of the satellite just before the collision?

(b)

What is the ratio of this kinetic energy to the kinetic energy of a

5.0 g

bullet from a modern army rifle with a muzzle speed of

950 m s^{- 1}

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Principles Of Physics International Student Version>Chapter 13 - Gravitation>Problems>Q 41

In $1993$ , the spacecraft Galileo sent an image of asteroid $243 Ida$ and a tiny orbiting Moon (now known as Dactyl), the first confirmed example of an asteroid-Moon system. In the image, the moon, which is $1.5 km$ wide, is $100 km$ away from the centre of the asteroid, which is $55 km$ long. Assume the Moon's orbit is circular with a period of $27 h$ . $(a)$ What is the mass of the asteroid? $(b)$ The volume of the asteroid, measured from the Galileo images, is $14100 {km}^{3} .$ What is the density (mass per unit volume) of the asteroid?

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Principles Of Physics International Student Version>Chapter 13 - Gravitation>Problems>Q 42

In the figure shown, not to scale, a cross-section through the interior of the Earth rather than being uniform throughout, the Earth is divided into three zones: an outer crust, a mantle, and an inner core. The dimensions of these zones and the masses contained within them are shown in the figure. The Earth has a total mass of $5.98 \times 10^{24} kg$ and a radius of $6370 km .$ Ignore rotation and assume that Earth is spherical.
$(a)$ Calculate $a_{g}$ at the surface.
$(b)$ Suppose that a borehole (the mole) is driven to the crust-mantle interface at a depth of $25.0 km;$ what would be the value of $a_{g}$ at the bottom of the hole?
$(c)$ Suppose that the Earth were a uniform sphere, with the same total mass and size. What would be the value of $a_{g}$ at a depth of $25.0 km ?$ (Precise measurements of $a_{g}$ are sensitive probes of the interior structure of the Earth, although results can be clouded by local variations in mass distribution).

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Principles Of Physics International Student Version>Chapter 13 - Gravitation>Problems>Q 43

In the figure shown below, the scale on which the $80 kg$ physicist stands reads $205 N$ . How long will the cantaloupe take to reach the floor if the physicist drops it (from rest relative to himself) at a height of $3.0 m$ above the floor?

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Principles Of Physics International Student Version>Chapter 13 - Gravitation>Problems>Q 44

Moon effect: Some people believe that the Moon controls their activities. If the Moon moves from being directly on the opposite side of the Earth from you, to being directly overhead, by what percent does

(a)

the Moon's gravitational pull on you increase and

(b)

your weight (as measured on a scale) decrease? Assume that the Earth-Moon (centre-to-centre) distance is

3.82 \times 10^{8} m

and Earth's radius is

6.37 \times 10^{6} m

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Principles Of Physics International Student Version>Chapter 13 - Gravitation>Problems>Q 45

Consider an object of mass

M

that is small enough to be treated as a particle but large enough, so that a piece of mass

m

is taken out from it and placed at

3.00 mm

away from it. What ratio of

\frac{m}{M}

gives the least gravitational potential energy for the system?

At what altitude above the Earth's surface would the gravitational acceleration be 2.0 m s-2?

Important Questions on Gravitation

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At what altitude above the Earth's surface would the gravitational acceleration be $2 .0 m s^{- 2}$ ?