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Classical Dynamics: Example Sheet 3

Michaelmas 2012Comments welcome: please send them to Berry Groisman (bg268@)

1. Tensor of Inertia

a. Prove that the principal moments of inertia, Ia, are real and non-negative.

b. During the lectures we have outlined the proof of the Parallel Axis Theorem, which

is a statement that the inertia tensor about a point P, which is displaced by c from

the centre of mass is related to the inertia tensor about the centre of mass as

(IP

)ab = (ICoM)ab + M(c

2

ab cacb), (1)where M is the total mass of the body. Complete the proof of the theorem. Hint: it

will be helpful to choose the origin in the centre of mass.

2. Show that the effect of three rotations by Euler angles results in the relationship

ea = Rabeb between the body frame axes {ea} and the space frame axes {e} where the

orthogonal matrix R is

R =

cos cos cos sin sin sin cos + cos sin cos sin sin

cos sin cos cos sin sin sin + cos cos cos sin cos

sin sin sin cos cos

Use this to confirm that the angular velocity can be expressed in terms of Euler

angles as

= [ sin sin + cos ]e1 + [ sin cos sin ]e2 + [ + cos ]e3 (2)

in the body frame {ea}. Or, alternatively, as

= [ sin sin + cos ]e1 + [ sin cos + sin ]e2 + [ + cos ]e3 (3)

in the space frame {ea}.

3. The physicist Richard Feynman tells the following story:

I was in the cafeteria and some guy, fooling around, throws a plate in the

air. As the plate went up in the air I saw it wobble, and I noticed the red

medallion of Cornell on the plate going around. It was pretty obvious to me

that the medallion went around faster than the wobbling.

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I had nothing to do, so I start figuring out the motion of the rotating plate.

I discover that when the angle is very slight, the medallion rotates twice as

fast as the wobble rate two to one. It came out of a complicated equation!I went on to work out equations for wobbles. Then I thought about how the

electron orbits start to move in relativity. Then theres the Dirac equation

in electrodynamics. And then quantum electrodynamics. And before I knew

it....the whole business that I got the Nobel prize for came from that piddling

around with the wobbling plate.

Feynman was right about quantum electrodynamics. But what about the plate?

4. Consider a heavy symmetric top of mass M, pinned at point P which is a distance

Figure 1: The Euler angles for the heavy symmetric top

l from the centre of mass. The principal moments of inertia about P are I1, I1 and I3and the Euler angles are shown in the figure. The top is spun with initial conditions

= 0 and = 0. Show that obeys the equation of motion,

I1 = Veff()

(4)

where

Veff() =I23

2

3

2I1

(cos cos 0)2

sin2 + Mgl cos (5)

Suppose that the top is spinning very fast so that

I33

MglI1 (6)

Show that 0 is close to the minimum of Veff(). Use this fact to deduce that the top

nutates with frequency

3I3

I1(7)

and draw the subsequent motion.

5. Throw a book in the air. If the principal moments of inertia are I1 > I2 > I3,

convince yourself that the book can rotate in a stable manner about the principal axes

e1 and e3, but not about e2.

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Use Eulers equations to show that the energy E and the total angular momentum

L2 are conserved. Suppose that the initial conditions are such that

L2 = 2I2E (8)

with the initial angular velocity perpendicular to the intermediate principal axes e2.

Show that will ultimately end up parallel to e2 and derive the characteristic time

taken to reach this steady state.

6. A rigid lamina (i.e. a two dimensional object) has principal moments of inertia

about the centre of mass given by,

I1 = (2 1) I2 = (

2 + 1) , I3 = 22 (9)

Write down Eulers equations for the lamina moving freely in space. Show that the

component of the angular velocity in the plane of the lamina (i.e.

21 + 22) is constant

in time.

Choose the initial angular velocity to be = Ne1 + Ne3. Define tan = 2/1,

which is the angle the component of in the plane of the lamina makes with e1. Show

that it satisfies

+ N2 cos sin = 0 (10)

and deduce that at time t,

= [Nsech Nt]e1 + [Ntanh Nt]e2 + [Nsech Nt]e3 (11)

7. The Lagrangian for the heavy symmetric top is

L = 12

I1

2 + 2 sin2

+ 12

I3( + cos )2 Mgl cos (12)

Obtain the momenta p, p and p and the Hamiltonian H(,,,p, p, p). Derive

Hamiltons equations.

8. A system with two degrees of freedom x and y has the Lagrangian,

L = xy + yx2 + xy (13)

Derive Lagranges equations. Obtain the Hamiltonian H(x,y,px, py). Derive Hamil-

tons equations and show that they are equivalent to Lagranges equations.

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