42 Classical mechanics
inthe body-fixed frame. A similar relation can be derived for the time derivative of
the positionr 2 of atom 2. Indeed, eqn. (1.11.18) applies to the time derivative of any
arbitrary vectorG: (
dG
dt
)
space=
(
dG
dt)
body+ω×G. (1.11.19)Although it is possible to obtain eqn. (1.11.19) from a general consideration of rota-
tional motion, we content ourselves here with this physically motivated approach.
Consider, next, the force term−∂U/∂θ. This is also a component of a vector
quantity known as thetorqueabout thez-axis. In general,τis defined by
τ=r×F. (1.11.20)Again, because the motion is entirely in thexy-plane, there is noz-component of the
force, and the only nonvanishing component of the torque is thez-component given
by
τz=xFy−yFx=−dcosθ∂U
∂y
+dsinθ∂U
∂x=−dcosθ∂U
∂θ∂θ
∂y+dsinθ∂U
∂θ∂θ
∂x, (1.11.21)
where the chain rule has been used in the last line. Sinceθ= tan−^1 (y/x), the two
derivatives ofθcan be worked out as
∂θ
∂y=
1
1 + (y/x)^21
x=
x
x^2 +y^2=
cosθ
d∂θ
∂x=
1
1 + (y/x)^2(
−
y
x^2)
=−
y
x^2 +y^2=−
sinθ
d. (1.11.22)
Substitution of eqn. (1.11.22) into eqn. (1.11.21) gives
τz=−∂U
∂θ(
dcosθ
cosθ
d+dsinθ
sinθ
d)
=−
∂U
∂θ(
cos^2 θ+ sin^2 θ