hi shawli!
shawli said:
Oh, to add -- the back of the book says that L is *not* conserved since the bearing of the turn table exerts a force northward onto the turntable. I don't understand where that comes from at all, could someone please clarify?
never mind whether this is right or wrong … it seems to me to make no sense anyway …
a "force northward" doesn't necessarily create a torque (it depends whether it acts on or off the axis), so it doesn't necessarily affect L
also, since the axle is
frictionless, the external force can only be perpendicular to the surface of the axle, ie along the radius and through the centre of the axle, and so exerts no torque about the axle
(
two external forces along different lines of action can of course constitute a torque)
shawli said:
Is the angular momentum of the system constant?
since I'm disagreeing with the answer in the book, i'll take the unusual step of giving a full answer …
Euler's equations (for a frame of reference fixed in the body along three (perpendicular) principal axes, and therefore rotating with it) are:
\tau_1\ =\ I_1\,\frac{d\omega_1}{dt} + (I_3\ -\ I_2)\omega_2\omega_3
\tau_2\ =\ I_2\,\frac{d\omega_2}{dt} + (I_1\ -\ I_3)\omega_3\omega_1
\tau_3\ =\ I_3\,\frac{d\omega_3}{dt} + (I_2\ -\ I_1)\omega_1\omega_2
we're only interested in the component along the axis of the turntable (say, the third equation), and since presumably I
2 - I
1 = 0, that reduces to (for the turntable only):
\tau_3\ =\ I_3\,\frac{d\omega_3}{dt}
the component of the turntable's angular momentum parallel to the axis of the turntable
caused by the Earth's rotation is constant, so it does not affect \omega_3, so the third equation is the same whether the Earth is rotating or not
so the component of external torque on the turntable parallel to its axis is the same as if the Earth was not rotating, and is equal and opposite to the component of the external torque on the woman, and the total angular momentum parallel to the axis
is conserved
(there's probably a simpler way of doing this that doesn't involve Euler's equations … but it's just after midnight here, and i don't see it! :zzz:)