Begin2.DVI

9-17. (Research project on orbital motion)

Assume a mass m located at a position
r =reˆr experiences a central force

F
=mf (r)ˆer

(a) Show the equation of motion is given by md

(^2)
r
dt^2

=F
which can be written in the

form d
v

dt

=f(r)ˆer

(b) Show that
r ×
v =
his a constant.

(c) Show the motion of mis in a plane and that the mass m sweeps out an area at

a constant rate. (Kepler’s law of areas).

(d) Show that in the special case mf (r) = −

GmM

r^2 the mass mis attracted toward

mass M, assumed to be at the origin, and

d
v

dt = −

k

r^2

ˆer where k =GM is a

constant.

(e) Show that
h=r^2 ˆer×d

ˆer

dt ,

d
v

dt ×

h=kdˆer

dt and
v ×

h=k(ˆer+

)where

is a constant

vector.

(f) Use the results from part (e) and show

r ×
v ·
h=h^2 and
r ·
v ×
h=kr (1 + cos θ)

where θis the angle between

and
r , and consequently

r=

α

1 + cos θ, where α=

h^2

k

Note rdescribes a conic section having eccentricity.

(f) When < 1 show an ellipse results with mhaving an orbital period

T=

area of ellipse

h/ 2 =

√^2 π

k

a^3 /^2 , where

T^2

a^3 =

4 π^2

k

This is known as Kepler’s third law.

9-18. (a) Find the potential associated with the conservative vector field

F
= (y^2 cos x+z^3 )ˆe 1 + (2 ysin x−4) ˆe 2 + 3 xz^2 ˆe 3

(b) Find the differential equation which describes the field lines.

9-19. Show that the vector field

F
 = (2xyz +y)ˆe 1 + (x^2 z+x)ˆe 2 +x^2 yeˆ 3

is conservative and find its scalar potential.