T Shift

| Problems

13. (II) Suppose the mass of the Earth were doubled, by

6-1 to 6-3 Law of Universal Gravitation

kept the same density and spherical shape. How would .

21. (Ill) De

weight of objects at the Earth's surface change?

live valu

14 (II) Given that the acceleration of gravity at the surface

Earth is

1. (1) Calculate the force of Earth's gravity on a spacecraft

2:00 Earth radii above the Earth's surface if its mass is 1480 kg.

- 22. (III) It

2. (1) Calculate the acceleration due to gravity on the Moon.

Mars is 0.38 of what it is on Earth, and that Mars' radio

3400 km, determine the mass of Mars.

the force

The Moon's radius is 1.74 x 10'm and its mass is

on the

735 x 10* kg.

IS. (II) At what distance from the Earth will a spacecraft or

Earth, but has the same mass What is the acceleration due

cling directly from the Earth to the Moon experience pek

of grav

A. (1) A hypothetical planet has a radius 23 times that of

net force because the Earth and Moon pull with equal s

cancel

reach

to gravity near its surface?

opposite forces?

16. (II) Determine the mass of the Sun using the known vaj

Appro

4 (1) A hypothetical planet has a mass 180 times that of

Earth, but the same radius What is & near its surface?

(1) If you doubled the mass and tripled the radius of a

for the period of the Earth and its distance from the gin

6-4 Sat

planet, by what factor would g at its surface change?

Ifhint: The force on the Earth due to the Sun is related

6. (II) Calculate the effective value of g. the acceleration of gravity,

the centripetal acceleration of the Earth.] Compare you

23. (1) Th

680 K

movin

at (a) 6400m, and (6) 6400 km, above the Earth's surface

answer to that obtained using Kepler's laws, Example 6

17. (II) Two identical point masses, each of mass M, alway

remain separated by a distance of 2R. A third mass at is g

24. (1)

7. (II) You are explaining to friends why astronauts feel

weightless orbiting in the space shuttle, and they respond that

they thought gravity was just a lot weaker up there, Convince

placed a distance x along the perpendicular bisector of The

Circu

25. (11)

bath

them and yourself that it isn't so by calculating how much

original two masses, as shown in Fig. 6-26. Show that The

weaker gravity is 300 km above the Earth's surface.

gravitational force on the third

15 1

& (II) Every few hundred years most of the planets line up on

mass is directed inward along

OM

the same side of the Sun. Calculate the total force on the Earth

the perpendicular bisector and

26. (11)

ceil

due to Venus, Jupiter, and Saturn, assuming all four planets

has a magnitude of

18

are in a line, Fig. 6-24. The masses are My = 0,815 ME.

My = 318 ME. Meat = 95.1 ME, and the mean distances

F =

2GM MIX

(x2 + RZ)

27

of the four planets from the Sun are 108, 150, 778, and

Earth is this?

1430 million km. What fraction of the Sun's force on the

FIGURE 6-26

Problem 17.

LOM

28.

Earth

Jupiter

Satur

18. (II) A mass M is ring shaped with radius r. A small mass m

Sun

is placed at a distance x along the ring's axis as shown in

29

Fig. 6-27. Show that the gravitational force on the mass m due

FIGURE 6-24 Problem 8 (not to scale).

to the ring is directed inward along the axis and has magnitude

GMmx

9. (II) Four 8.5-kg spheres are located at the corners of a square

30.

of side 0.80 m. Calculate the magnitude and direction of the

gravitational force exerted on one sphere by the other three.

[ Himr: Think of the ring as made up

10. (II) Two objects attract each other gravitationally with a

of many small point masses dM; sum

force of 2.5 x 10 N when they are 0.25 m apart. Their

over the forces due to each dM, and

total mass is 4.00 kg. Find their individual masses.

use symmetry.]

11. (II) Four masses are arranged as shown in Fig. 6-25.

Determine the x and y components of the gravitational

force on the mass at

FIGURE 6-27

the origin (m). Write

Problem 18.

the force in vector

19. (III) (a) Use the binomial expansion

notation (i, j).

(1 + x)" = 1+nxx +

XO

n(n - 1) 2 ...

2

to show that the value of g is altered by approximately

Ag =-28

Yo

FIGURE 6-25

Problem 11.

at a height Ar above the Earth's surface, where ry is the

radius of the Earth, as long as Ar < rE. (b) What is the

12. (II) Estimate the acceleration due to gravity at the surface

meaning of the minus sign in this relation? (c) Use this

of Europa (one of the moons of Jupiter) given that its mass

result to compute the effective value of g at 125 km above

is the same as Earth's.

is 4.9 x 10"kg and making the assumption that its density

the Earth's surface. Compare to a direct use of Eq. 6-1.

20. (III) The center of a 1.00 km diameter spherical pocket of of

is 1.00 km beneath the Earth's surface. Estimate by what

158 CHAPTER 6 Gravitation and Newton's Synthesis

percentage g directly above the pocket of oil would differ

from the expected value of g for a uniform Earth? Assume

the density of oil is 8.0 x 10- kg/m'.