Simple Nature - Light and Matter

angular momentum is related to rate at which area is swept out by the line segment connecting

the particle to the axis.

Torqueis the rate of change of angular momentum,τ= dL/dt. The torque created by a

given force can be calculated using any of the relations

τ=rFsinθrF

=rF⊥

=r⊥F,

where the subscript⊥indicates a component perpendicular to the line connecting the axis to

the point of application of the force.

In the special case of arigid bodyrotating in a single plane, we define

ω=

dθ

dt

[angular velocity]

and

α=

dω

dt

, [angular acceleration]

in terms of which we have

L=Iω

and

τ=Iα,

where themoment of inertia,I, is defined as

I=

∑

miri^2 ,

summing over all the atoms in the object (or using calculus to perform a continuous sum, i.e.

an integral). The relationship between the angular quantities and the linear ones is

vt=ωr [tangential velocity of a point]

vr= 0 [radial velocity of a point]

at=αr. [radial acceleration of a point]

at a distancerfrom the axis]

ar=ω^2 r [radial acceleration of a point]

at a distancerfrom the axis]

In three dimensions, torque and angular momentum are vectors, and are expressed in terms of

the vectorcross product, which is the only rotationally invariant way of defining a multiplication

of two vectors that produces a third vector:

L=r×p

τ=r×F

1076 Chapter Appendix 5: Summary