218
5 Gravitation
5.31 For an elliptic orbit
v =
G M
2
r
−
1
a
∴ v max =
G M
2
r min
−
1
a
=
G M(2a − r min )
ar min
=
G Mr max
ar min
(1)
as r max + r min = 2a
v min =
G M
2
r max
−
1
a
=
G M(2a − r max )
ar max
=
G Mr min
ar max
(2)
Multiplying (1) and (2)
v max v min =
G M
a
or
√ v max v min =
G M
a
= a
G M
a 3 =
2πa
T
as T = 2π
a 3
G M
∴ a =
T
2π
√
v max v min
5.32
T = 2π
r 3
G M
(1)
But r = a + b + c
(2)
Combining (1) and (2)
M =
4π 2 (a + b + c) 3
GT 2
5.33 L = |r × p| = r p sin θ
L per unit mass = r v sin θ
= (1.75 × 1.5 × 10
11
)(3 × 10
4
) sin 30
◦
= 3.9375 × 10
15 m
2
/s
When the comet is closest to the sun its velocity will be perpendicular to
the radius vector. The angular momentum L = r v . Angular momentum
conservation requires
5 Gravitation
5.31 For an elliptic orbit
v =
G M
2
r
−
1
a
∴ v max =
G M
2
r min
−
1
a
=
G M(2a − r min )
ar min
=
G Mr max
ar min
(1)
as r max + r min = 2a
v min =
G M
2
r max
−
1
a
=
G M(2a − r max )
ar max
=
G Mr min
ar max
(2)
Multiplying (1) and (2)
v max v min =
G M
a
or
√ v max v min =
G M
a
= a
G M
a 3 =
2πa
T
as T = 2π
a 3
G M
∴ a =
T
2π
√
v max v min
5.32
T = 2π
r 3
G M
(1)
But r = a + b + c
(2)
Combining (1) and (2)
M =
4π 2 (a + b + c) 3
GT 2
5.33 L = |r × p| = r p sin θ
L per unit mass = r v sin θ
= (1.75 × 1.5 × 10
11
)(3 × 10
4
) sin 30
◦
= 3.9375 × 10
15 m
2
/s
When the comet is closest to the sun its velocity will be perpendicular to
the radius vector. The angular momentum L = r v . Angular momentum
conservation requires