Chapter 11 Gravity

Gravity
expressing the mass of the sphere in terms of C. We can then use the definition of
the gravitational field to find the gravitational field of the sphere inside its surface.
The work done by the field equals the negative of the change in the potential
energy of the system as the small object falls in the hole.
Use conservation of energy to relate
the work done by the gravitational
field to the speed of the small object
as it strikes the bottom of the hole:
Express the mass of a differential
element of the sphere:
Integrate to express the mass of
K f − Ki
+ ΔU
= 0
or, because Ki = 0 and W = −ΔU,
1 2 2W
W = mv 2 ⇒ v = (1)
m
where v is the speed with which the
object strikes the bottom of the hole
and W is the work done by the
gravitational field.
dm = ρ dV = ρ
2 ( 4π r dr)
2
M 4 π C rdr = ( 50m
) π C
the sphere in terms of C:
Solving for C yields:
Substitute numerical values and
5.
0
= ∫
0
m
M
C = 2 ( 50m
)π
1.
0×
10
C =
evaluate C: 2 ( 50m
)
11
kg
8
= 6.
37×
10 kg/m
π
Use its definition to express the gravitational field of the sphere at a distance from
its center less than its radius:
2
4π
r ρ dr
Fg
GM 0
g = = = G = G
2
2
m r r
Express the work done on the small
object by the gravitational force
acting on it:
Substitute in equation (1) and
simplify to obtain:
r
∫
r
∫
0
4π
r
r
2
2
r
C
dr 4π
C r dr
r ∫
0
= G = 2π
GC
2
r
3.0m
W = − ∫ mgdr = ( 2m)mg
v =
=
5.
0
2
m
( 2.
0m)
m(
2π
GC)
m
( 8.
0m)
π GC
2
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