Sample Quiz 1, Physics 5011, Fall, 2007

Instructions: This is a closed book exam. Calculators are not permitted (note that any numerical answers required can be done to order-of-magnitude accuracy). Do each problem on the sheets of paper provided. Make sure your name and ID number are on every sheet, so they can be identified if they become separated. After you are done, staple together all of your sheets. 1. (30 pts.) A sphere of mass m and radius a rolls without slipping inside a spherical shell of

radius R (with R > a) under the influence of a uniform gravitational force. Assuming that all of the motion occurs in one vertical plane, set up appropriate generalized coordinates for this problem and determine the Lagrangian and any constraint conditions. Determine the equations of motion and find the frequency of small oscillations for this system. (Hint: recall that I = (2/5)ma2 for a sphere.)

2. (30 pts.) Consider a particle of mass m and angular momentum l that moves in a central force field that has the form F = –mkr. (a) Find the total energy and sketch the effective potential for this particle, and determine the radius and period of a circular orbit. (b) Suppose that the orbit is almost circular. Show that the particle reaches its closest distance to the center twice per orbit (or equivalently, that the frequency of small oscillations about the circular orbit is twice the orbital frequency).

3. (40 pts.) Consider a symmetrical top (I1 = I2 ≠ I3) with no torques acting upon it. (a) Write

down the Lagrangian for this top in terms of the Euler angles and determine two constants of motion associated with the generalized momenta of φ and ψ. Find the equation of motion for θ, and show that steady precession (constant θ) is possible when 3 3 1/ cosI Iϕ = ω θ . (b) Show that one of the two constants of motion in part (a) corresponds to the angular momentum around the body z′ axis and the other corresponds to the fixed z axis. (Hint: Determine the angular momentum vector in body coordinates, and perform the coordinate transformation to find the z component in the fixed coordinates.)

Physics 5011, Quiz 1, October 6, 2006 Important formulas: (Note that not all formulas are valid in all cases. Also, tensors are denoted by a double arrow above the symbol rather than a double underline) 21

2T mv m V V d= = = × = −∇ = − ⋅∫p v L r p F F r

i i i i

i ii cm cm

i

m mM m

M M= = =

∑ ∑∑

r vR V

Constraints: 0i ij ij ji jj j

d L LL T V a a dqdt q q⎛ ⎞∂ ∂

= − − = λ =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠∑ ∑

2j j j

jj

Lp h p q L l mrq∂

= = − = θ∂ ∑

( ) ( )2

2 2 2 21dV ud u m mu f u u

du rd l u l+ = − = − =

θ

[ ]2 2

max min max min2 2 2

max min

/ 21 cos( ) 12 2 1

r r r rmk k l mk Elu e a eE r rl e mk

+ −′= + θ−θ = = − = = = ++−

cot2 2 sink b dbbE d

Θ= σ =

Θ Θ

( )1 1

1

i ij j ji ik jk ijij

ij ik jl kl

v A v A A A

B A A B

− −

−

′ ′= ⋅ = = = = δ

′ ′= ⋅ ⋅ =

v A v A A A

B A B A

cos cos cos sin sin cos sin cos cos sin sin sinsin cos cos sin cos sin sin cos cos cos cos sin

sin sin sin cos cos

ψ ϕ− θ ϕ ψ ψ ϕ+ θ ϕ ψ ψ θ⎛ ⎞⎜ ⎟= − ψ ϕ− θ ϕ ψ − ψ ϕ+ θ ϕ ψ ψ θ⎜ ⎟⎜ ⎟θ ϕ − θ ϕ θ⎝ ⎠

A

1

cos cos cos sin sin sin cos cos sin cos sin sincos sin cos cos sin sin sin cos cos cos sin cos

sin sin cos sin cos

−

ψ ϕ− θ ϕ ψ − ψ ϕ− θ ϕ ψ θ ϕ⎛ ⎞⎜ ⎟= ψ ϕ+ θ ϕ ψ − ψ ϕ+ θ ϕ ψ − θ ϕ⎜ ⎟⎜ ⎟ψ θ ψ θ θ⎝ ⎠

A

sin sin cos cos sin sinsin cos sin sin sin coscos cos

x x

y y

z z

′

′

′

ω = ϕ θ ψ + θ ψ ω = θ ϕ+ψ θ ϕω = ϕ θ ψ −θ ψ ω = θ ϕ−ψ θ ϕω = ϕ θ+ψ ω = ψ θ+ϕ

( ) ( )fix bodyfix body

d ddt dt

⎛ ⎞ ⎛ ⎞= + × = + ×⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

u u uω ω

( )2eff m m= − × − × ×F F v rω ω ω ( )3 2 1

2d r r T⎡ ⎤= ρ − = ⋅ = ⋅ ⋅⎣ ⎦∫I r 1 rr L I Iω ω ω

( )2cm cm cm cm i i ijk j k k iM R I I N= + − ω + ε ω ω =I I 1 R R

( ) ( ) ( )22 2 21 3

1 1sin cos cos2 2

L I I V= θ +ϕ θ + ψ +ϕ θ − θ Math:

Binomial Theorem: ( ) ( ) ( ) ( )21 1 ... 11 1 ... ...

2 !p np p p p p n

x px x xn

− − − ++ = + + + + +

Taylor expansion: ( ) ( ) ( ) ( ) ( ) ( )2

0 0 0 ... 0 ...2 !

nnx xf x f xf f f

n′ ′′= + + + + +

Divergence Theorem: 3 ˆd x da∇⋅ = ⋅∫ ∫A n A Stokes Theorem: ( )ˆda d⋅ ∇× = ⋅∫ ∫n A A l

2 2 2 2 2 2sin cos 1 sec tan 1 cosh sinh 1θ+ θ = θ− θ = θ− θ = ( ) ( )2 21 1cos 1 cos 2 sin 1 cos 2 sin 2 2sin cos

2 2θ = + θ θ = − θ θ = θ θ

1 for 123, 231,3121 for

1 for 132, 213,3210 for

0 otherwiseijk ij ijk ilm jl km jm kl

ijki j

ijki j

=⎧=⎧⎪ε = − = δ = ε ε = δ δ − δ δ⎨ ⎨ ≠⎩⎪

⎩( ) ( ) ( ) ( ) ( ) ( )× × = ⋅ − ⋅ ⋅ × = ⋅ × = ⋅ ×A B C B A C C A B A B C B C A C A B

( ) ( ) ( )( ) ( )( )× ⋅ × = ⋅ ⋅ − ⋅ ⋅A B C D A C B D A D B C

( ) ( ) ( ) 20 0∇×∇ψ = ∇⋅ ∇× = ∇× ∇× = ∇ ∇⋅ −∇A A A A ( ) ( )∇ ⋅ ψ = ∇ψ ⋅ +ψ∇⋅ ∇× ψ = ∇ψ× +ψ∇×A A A A A A ( ) ( ) ( )∇ ⋅ = ⋅∇ + ⋅∇ + × ∇× + × ∇×A B A B B A A B B A ( ) ( ) ( ) ( )∇ ⋅ × = ⋅ ∇× − ⋅ ∇× ∇× × = ∇ ⋅ − ∇ ⋅ + ⋅∇ − ⋅∇A B B A A B A B A B B A B A A B Div, Grad, Curl and all that: Cylindrical Spherical

1ˆ ˆ ˆs s z

∂ψ ∂ψ ∂ψ∇ψ = + +

∂ ∂ϕ ∂s zϕ 1 1ˆˆ ˆ

sinr r r∂ψ ∂ψ ∂ψ

∇ψ = + +∂ ∂θ θ ∂ϕ

r θ ϕ

( )1 1 zs

A AsAs s s z

ϕ∂ ∂∂∇ ⋅ = + +

∂ ∂ϕ ∂A ( ) ( )2

21 1 1sin

sin sinrA

r A Ar r rr

ϕθ

∂∂ ∂∇ ⋅ = + θ +

∂ θ ∂θ θ ∂ϕA

( )

1ˆ ˆ

1ˆ

sz z

s

A AA As z z s

AsAs s

ϕ

ϕ

∂⎛ ⎞ ∂∂ ∂⎛ ⎞∇× = − + −⎜ ⎟ ⎜ ⎟∂ϕ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎡ ⎤∂∂

+ −⎢ ⎥∂ ∂ϕ⎣ ⎦

A s

z

ϕ ( )

( ) ( )

1ˆ sinsin

1 1 1ˆ ˆsin

r r

AAr

A ArA rAr r r r r

θϕ

ϕ θ

⎡ ⎤∂∂∇× = θ −⎢ ⎥θ ∂θ ∂ϕ⎣ ⎦

⎡ ⎤∂ ∂∂ ∂⎡ ⎤+ − + −⎢ ⎥ ⎢ ⎥θ ∂ϕ ∂ ∂ ∂θ⎣ ⎦⎣ ⎦

A r

θ ϕ

2 22

2 2 21 1ss s s s z∂ ∂ψ ∂ ψ ∂ ψ⎛ ⎞∇ ψ = + +⎜ ⎟∂ ∂ ∂ϕ ∂⎝ ⎠

( )2 2

22 2 2 2 2

1 1 1sinsin sin

rr r r r∂ ∂ ∂ψ ∂ ψ⎛ ⎞∇ ψ = ψ + θ +⎜ ⎟∂θ ∂θ∂ θ θ ∂ ϕ⎝ ⎠

2 2 2 2 2dl ds s d dz= + ϕ + 2 2 2 2 2 2 2sindl dr r d r d= + θ + θ ϕ 3d x s ds d dz= ϕ ( )3 2 2sin cosd x r dr d d r dr d d= θ θ ϕ = θ ϕ

ˆ ˆ ˆcos sinˆ ˆ ˆsin cosˆ ˆ

= ϕ + ϕϕ = − ϕ + ϕ=

s x yx y

z z

ˆ ˆ ˆ ˆsin cos sin sin cosˆ ˆ ˆ ˆcos cos cos sin sinˆ ˆ ˆsin cos

= θ ϕ + θ ϕ + θ

θ ϕ + θ ϕ − θ− ϕ + ϕ

r x y z

x y zx y

θ =ϕ =