5.3 Solutions
201
5.48 Find the law of force for the orbit r = a sin nθ .
5.49 Find the law of force to the pole when the orbit described by the cardioid
r = a(1 − cos θ).
5.50 In prob. (5.49) prove that if Q be the force at the apse and v the velocity,
3v 2 = 4a Q.
5.51 A particle moves in a plane under an attractive force varying as the inverse
cube of the distance. Find the equation of the orbit distinguishing three cases
which may arise.
5.52 Show that the central force necessary to make a particle describe the lemniscate r 2 = a 2 cos 2θ is inversely proportional to r 7 .
5.53 Show that if a particle describes a circular orbit under the influence of an
attractive central force directed towards a point on the circle, then the force
varies as the inverse fifth power of distance.
5.54 If the sun’s mass suddenly decreased to half its value, show that the earth’s
orbit assumed to be originally circular would become parabolic.
5.3 Solutions
5.3.1 Field and Potential
5.1 F =
G M 1 M 2
r 2
If R is the radius of either sphere, the distance between the centre of the spheres
in contact is r = 2R:
M 1 = M 2 = M =
4
3
π R
3
ρ
F =
G M 2
4R 2 =
4π 2 G R 4 ρ 2
9
=
4π 2
9
× 6.67 × 10
−11
× (0.2)
4
(11300)
2
= 5.98 × 10
−5 N
5.2 As the mass of A and B are identical and the distance PA = PB, the magnitude
of the force F PA = F PB . Resolve these forces in the horizontal and vertical
direction. The horizontal components being in opposite direction get cancelled.
The vertical components get added up.
F PA = F PB = G
(0.01)(0.26)
(0.1) 2
= 0.26G
201
5.48 Find the law of force for the orbit r = a sin nθ .
5.49 Find the law of force to the pole when the orbit described by the cardioid
r = a(1 − cos θ).
5.50 In prob. (5.49) prove that if Q be the force at the apse and v the velocity,
3v 2 = 4a Q.
5.51 A particle moves in a plane under an attractive force varying as the inverse
cube of the distance. Find the equation of the orbit distinguishing three cases
which may arise.
5.52 Show that the central force necessary to make a particle describe the lemniscate r 2 = a 2 cos 2θ is inversely proportional to r 7 .
5.53 Show that if a particle describes a circular orbit under the influence of an
attractive central force directed towards a point on the circle, then the force
varies as the inverse fifth power of distance.
5.54 If the sun’s mass suddenly decreased to half its value, show that the earth’s
orbit assumed to be originally circular would become parabolic.
5.3 Solutions
5.3.1 Field and Potential
5.1 F =
G M 1 M 2
r 2
If R is the radius of either sphere, the distance between the centre of the spheres
in contact is r = 2R:
M 1 = M 2 = M =
4
3
π R
3
ρ
F =
G M 2
4R 2 =
4π 2 G R 4 ρ 2
9
=
4π 2
9
× 6.67 × 10
−11
× (0.2)
4
(11300)
2
= 5.98 × 10
−5 N
5.2 As the mass of A and B are identical and the distance PA = PB, the magnitude
of the force F PA = F PB . Resolve these forces in the horizontal and vertical
direction. The horizontal components being in opposite direction get cancelled.
The vertical components get added up.
F PA = F PB = G
(0.01)(0.26)
(0.1) 2
= 0.26G