230
5 Gravitation
2 p
d p
dr
=
3r 2
2a
(5)
Using (5) in (4)
f = −
3
4
h 2 r 2
ap 4 = −
3ah 2
r 4
(6)
where we have used (3). Thus the force is proportional to the inverse fourth
power of distance.
5.50 If u =
1
r
, then at the apse
du
d θ
= 0
or −
1
r 2
dr
d θ
= 0
∴ −
1
r 2 a sin θ = 0
from which either sin θ = 0 or r is infinite, the latter case being inadmissible
so long the particle is moving along the cardioid.
Thus θ = π or 0
When θ = π, r = 2a
and Q =
3ah 2
r 4 =
3ah 2
16a 4 =
3h 2
16a 3
Also p
2
=
r 2
2a
=
8a 3
2a
= 4a
2
and v
2
=
h 2
p 2 =
h 2
4a 2
Thus 4a Q =
3h 2
4a 2 = 3v
2
When θ = 0, r = 0 and p = 0 and the particle is moving with infinite velocity
along the axis of the cardioid and continues to move in a straight line.
5.51 Let the force f = −
k
r 3 = −ku 3
where u =
1
r
d 2 u
d θ 2 + u = −
f
h 2 u 2 =
ku
h 2
5 Gravitation
2 p
d p
dr
=
3r 2
2a
(5)
Using (5) in (4)
f = −
3
4
h 2 r 2
ap 4 = −
3ah 2
r 4
(6)
where we have used (3). Thus the force is proportional to the inverse fourth
power of distance.
5.50 If u =
1
r
, then at the apse
du
d θ
= 0
or −
1
r 2
dr
d θ
= 0
∴ −
1
r 2 a sin θ = 0
from which either sin θ = 0 or r is infinite, the latter case being inadmissible
so long the particle is moving along the cardioid.
Thus θ = π or 0
When θ = π, r = 2a
and Q =
3ah 2
r 4 =
3ah 2
16a 4 =
3h 2
16a 3
Also p
2
=
r 2
2a
=
8a 3
2a
= 4a
2
and v
2
=
h 2
p 2 =
h 2
4a 2
Thus 4a Q =
3h 2
4a 2 = 3v
2
When θ = 0, r = 0 and p = 0 and the particle is moving with infinite velocity
along the axis of the cardioid and continues to move in a straight line.
5.51 Let the force f = −
k
r 3 = −ku 3
where u =
1
r
d 2 u
d θ 2 + u = −
f
h 2 u 2 =
ku
h 2