Chapter 11 Gravity

Gravity
Picture the Problem We can determine the maximum range at which an object
with a given mass can be detected by substituting the equation for the
gravitational field in the expression for the resolution of the meter and solving for
the distance. Differentiating g(r) with respect to r, separating variables to obtain
dg/g, and approximating Δr with dr will allow us to determine the vertical change
in the position of the gravity meter in Earth’s gravitational field is detectable.
(a) Earth’s gravitational field is
given by:
Express the gravitational field due to
the mass m (assumed to be a point
mass) of your friend and relate it to
the resolution of the meter:
Solving for r yields:
GM
g E =
R
g
() r
r = R
E
2
E
Gm
= = 1.
00×
10
2
r
−11
GM
= 1.
00×
10
R
E
1.
00×
10
M
Substitute numerical values and
11
6 1.
00×
10
evaluate r: ( ) ( 80kg)
r = 6.37× 10 m
24
(b) Differentiate g(r) and simplify to
obtain:
=
7.
37m
E
11
m
−11
E
2
E
g
E
5.98×
10
kg
dg − 2Gm
2 ⎛ Gm ⎞ 2
= = − = − g
3 ⎜ 2 ⎟
dr r r ⎝ r ⎠ r
Separate variables to obtain: dg dr −11
= −2
= 10
g r
Approximating dr with Δr, evaluate
Δr with r = RE:
Δr
= −
1
2
−11
6
( 1.
00×
10 )( 6.
37×
10 m)
= 31.
9μm
43 •• Earth’s radius is 6370 km and the moon’s radius is
1738 km. The acceleration of gravity at the surface of the moon is 1.62 m/s 2 .
What is the ratio of the average density of the moon to that of Earth?
Picture the Problem We can use the definitions of the gravitational fields at the
surfaces of Earth and the moon to express the accelerations due to gravity at these
locations in terms of the average densities of Earth and the moon. Expressing the
ratio of these accelerations will lead us to the ratio of the densities.
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