Description
Question 1 (3 marks)
In class we expressed the components of the angular velocity ω along the body-fixed axes in terms of Euler
angles as
ωbf =
φ˙ sin θ sin ψ + ˙θ cos ψ
φ˙ sin θ cos ψ − ˙θ sin ψ
φ˙ cos θ + ψ˙
. (1)
Show that the angular velocity along the space-fixed axes in terms of Euler angles is instead give by,
ωsf =
ψ˙ sin θ sin φ + ˙θ cos φ
−ψ˙ sin θ cos φ + ˙θ sin ψ
ψ˙ cos θ + φ˙
. (2)
Question 2 (2 marks)
Consider a 3D cone with uniform mass density (and total mass M).
(a) Find the moment of inertia tensor assuming the body-fixed co-ordinate system is such that the origin
is placed at:
i) the center-of-mass (COM) of the cone and the z-axis passes through the sharp tip of the cone
ii) the sharp tip (apex) of the cone and the z-axis passes through the COM
(b) Check your results to (a) by verifying that they are consistent with Steiner’s parallel axes theorem:
Ijk = I
0
jk − M(|R|
2
δjk − RjRk), (3)
where I is the moment of inertia tensor in the COM basis, I
0
relative to the cone tip and R is the
vector from the origin O of the COM basis to the cone tip.
Question 3 (3 marks)
Consider a toy model of a diatomic molecule where the atoms are taken to be point particles with masses
m1 and m2 connected by a massless rigid rod of length 2b.
Assume the molecule rotates in such a fashion
that the rigid rod makes a constant angle θ0 with respect to the z-axis of the space-fixed co-ordinate system
and the atoms trace out circular orbits in respective xy-planes (i.e., their space-fixed z co-ordinates are
fixed).
What is the angular momentum of the molecule and what is the magnitude of the torque that must
be applied (in the space-fixed frame) for the molecule to continue precessing about a fixed axis? Solve this
question by:
(a) computing the momentum of inertia using a body-fixed co-ordinate system set by the principle axes
(b) computing the rate of change of the angular momentum using a space-fixed co-ordinate system
Question 4 (2 marks)
Consider a cone of mass M, height h and half-angle α rolling on a plane without slipping. The cone is taken
to be orientated such that it is rolling on its slanted side (i.e., both the apex of the cone and edge of the
base make contact with the surface upon which the cone is rolling). Compute the kinetic energy.