202
5 Gravitation
Each vertical component = 0.26 G ×
6
10
= 0.156 G
Therefore F Net = 2×0.156 G = 2×0.156×6.67×10 −11 N = 2.08×10 −1 N
5.3 At distance r, F M = F m =
G Mm
r 2
Acceleration of mass m a m =
F m
m
=
G M
r 2
Acceleration of mass M a M =
F m
M
=
Gm
r 2
a rel = a m + a M =
G(M + m)
r 2
a rel =
dv rel
dt
= v rel
dv rel
dr
=
G(M + m)
r 2
Integrating
v rel dv rel =
v 2
rel
2
= G(M + m)
∞
d
dr
r 2 =
G(M + m)
d
∴
v rel =
2G(M + m)
d
5.4 Gravitational force F = −
G Mm
x 2
where M and m are the masses of the sun and the earth which are a distance x
apart.
Earth’s acceleration
a =
dv
dt
=
F
m
= −
G M
x 2
∴
dv
dt
=
vdv
dx
= −
G M
x 2
vdv = −G M
dx
x 2
Integrating
vdv =
v 2
2
= −Gm
dx
x 2 + C
where C = constant