3.4 Applications of the Vector Product 43
Fig. 3.1 Vector product
3.4 Applications of the Vector Product
3.4.1 Orbital Angular Momentum
Theorbital angular momentumLof a mass point at the positionrwith thelinear
momentumpis defined by
L=r×p. (3.43)
When the linear momentum is just massmtimes the velocityv,(3.43) is equivalent
toL=mr×v.
Notice: the orbital angular momentum depends on the choice of the origin of the
coordinate system wherer=0. Furthermore,Lis non-zero even for a motion along
a straight line, as long as the line does not go through the pointr=0. For constant
speedv, the magnitudeLofLis determined byr^0 mvwherer^0 is the shortest
distance of the line fromr=0, cf. Fig.3.2. The angular momentum is perpendicular
to the plane, pointing downward.
3.3 Exercise: Angular Momentum in Terms of Spherical Components
Compute thez-component of the angular momentum in terms of the spherical com-
ponents (2.10).
3.4.2 Torque
According to Newton, the time change of the linear momentum a particle, subjected
to a forceF, is determined by
dp
dt
:=p ̇=F. (3.44)