I7-4 (Continued)

κ^2 =dˆet

ds

·dˆet

ds

= α

(^2) ω 4
(α^2 ω^2 +β^2 )^2
Curvature κ= αω
2
α^2 ω^2 +β^2
radius of curvature ρ=
1
κ=
α^2 ω^2 +β^2
αω^2
unit normal eˆn=
1
κ
dˆet
ds =−cosωt
ˆe 1 −sinωtˆe 2
unit binormal ˆeb=ˆet׈en=βsinωtˆe 1 −βcosωtˆe 2 +
√ αω
α^2 ω^2 +β^2
ˆe 3
dˆeb
ds =
dˆeb
dt
ds
dt

βωcosωtˆe 1 +βωsinωteˆ 2
α^2 ω^2 +β^2 =−τ
ˆen
τ^2 =dˆeb
ds
·dˆeb
ds
= β
(^2) ω 2
(α^2 ω^2 +β^2 )^2
τ= βω
α^2 ω^2 +β^2
, σ=^1
τ
=α
(^2) ω (^2) +β 2
βω
I7-5.
~r′=d~rds
(
ds
dt
) 2
=d~rdt·d~rdt=~r′·~r′, dsdt= (~r′·~r′)^1 /^2
ˆet=d~r
ds

d~r
dt
ds
dt
, d
ˆet
dt

ds
dt
d^2 ~r
dt^2 −
d~r
dt
d^2 s
dt^2
(dsdt)^2
dˆet
ds =
dˆet
dt
ds
dt

~r′′
~r′·~r′−
~r′d
(^2) s
dt^2
~r′·~r′dsdt
=κˆen
= ~r
′′
~r′·~r′
−
~r′(~r′·~r′′)
(~r′·~r′)^1 /^2
~r′·~r′(~r′·~r′)^1 /^2
= ~r
′′
~r′·~r′
−~r
′(~r′·~r′′)
(~r′·~r′)^2
=κˆen
κ^2 =(κeˆn)·(κeˆn) =
(~r′·~r′)(~r′′·~r′′)−(~r′·~r′′)^2
(~r′·~r′)^3
I7-6. Special case of previous problem
~r=xeˆ 1 +y(x)ˆe 2 , ~r′=ˆe 1 +y′eˆ 2 , ~r′′=y′′eˆ 2
and
~r′·~r′= 1 + (y′)^2 , ~r′·~r′′=y′y′′, ~r′′·~r′′= (y′′)^2
giving
κ=
√
(1 + (y′)^2 )(y′′)^2 −(y′)^2 (y′′)^2
(1 + (y′)^2 )^3 /^2
=⇒ κ= |y
′′|
(1 + (y′)^2 )^3 /^2
Solutions Chapter 7