v (t) t + dt→ m (t + dt

(Dated: June 4, June 18, July 1, 2005)

Dynamics of systems of particles and rigid body

mi → rⁱ , i = 1,· · · , N mi

d²ri

dt² = Fi = Fi,ext+ XN

j=1 j6=i

Fij

→X

i

mi

d²ri

dt² =X

i

Fi,ext ∵ F^ij =−F^ji define:

rCM = 1 M

X

i

miri , M =X

i

mi , rCM: center of mass

→ Md²rCM

dt² =X

i

Fi,ext = Fext total external force total momentum

ptotal=X

i

mivi = MdrCM

dt = M vCM

∴ dptotal

dt = Fext

example: rocket t→ m (t) , v (t)

t + dt→ m (t + dt) = m (t) + dm (t) , v (t + dt) = v (t) + dv (t)

∴ p^total(t) = m (t) v (t) ptotal(t + dt) =

Ã

m (t) + dm|{z}

<0

!

(v (t) + dv) +|dm| (vrelative+ v (t))

| {z }

velocity of |dm| w.r.t. ground

∴ dptotal= ptotal(t + dt)− p^total(t) = Fextdt→ impulse

∴ mdv (t) − dmv^rel = Fextdt ; neglect the dm · dv term

∴ mdv

dt = Fext+ vrel

dm

| {z }dt

thrust

note: vrel//− v & dm dt < 0

total angular momentum Ltotal=X

i

mri× vⁱ =X

i

ri× pⁱ

ri = rCM+ r⁰_i ∴ vi = vCM+ v_i⁰ , r⁰_i, v_i⁰ → relative position & velocity of mⁱ w.r.t. CM

→X

i

mir⁰_i = 0 & X

i

miv_i⁰ = 0

→ L^total= M rCM × v^CM +X

i

mir_i⁰× v⁰i

= LCM + L⁰ → exists only if the system rotates w.r.t. an axis through CM with angular velocity

torque Γ Γ =X

i

Γi =X

i

ri× Fⁱ =X

i

(rCM + r_i⁰)× Fⁱ

= rCM ×X

i

Fi+ X

i

r⁰_i× Fⁱ

| {z }

torques about CM

= ΓCM + Γ⁰

∵ MdvCM

dt =X

i

Fi

∴ dLCM

dt = rCM ×X

i

Fi (refer to O) dLtotal

dt =X

i

mri× dvi

dt =X

i

ri× Fⁱ = rCM ×X

i

Fi+X

i

r_i⁰× Fⁱ

∴ dL⁰

dt =X

i

r_i⁰× Fⁱ (refer to CM)

Rigid body

rotation about its symmetric (principal) axis v_i⁰ = ω× ri⁰ & |ri⁰| = const. , I^CM =X

i

mi(r⁰_i⊥)² moment of inertia

L⁰ =X

i

mir⁰_i× vi⁰ =X

i

mir_i⁰× (ω × ri⁰) =X

i

mi(r_i⊥⁰ )²ω L⁰ = ICMω , ω angular velocity // the axis of rotation ICM

dω

dt = ICMα = Γ⁰ a) ring with ω // central axis

|ω| = dθ

dt , dθ = ωdt→ vector ICM = M R²

θ ^R

CM

ω^r

M

b) Disc with ω // central axis ICM = 1

2M R² c) Solid sphere

ICM = 2 5M R² d) Parallel axis theorem

I⁰ = ICM + M R²_//

R//

ω^r ω^r'

CM

note: in general L⁰ ∦ ω L⁰ =X

i

Lieˆi , Li =X

j

Iijωj , ω =X

j

ωjeˆj

→ inertia tensor

Rotational Dynamics: pure rolling

ICMα = Γ⁰, α≡ dω

dt angular acceleration

R

CM

A

B N r

f

r a r ,

v r

⊥

||

g β Mr

ω, α⊗

approach 1: refer to CM (noninertial reference)

ICM

dω

dt = r_{CM →A}× f^µ → I^CMα = Rfµ

M aCM = N + M g + fµ

//-comp. :b M aCM = M g sin β− f^µ

⊥-comp. : 0 = N − Mg cos βb

constraint condition aCM = α· R (∵ ∆rCM = R· ∆θ ∴ v^CM = R· ω)

∴ a^CM = M g sin β M +ICM

R²

approach 2: refer to the contact point A (inertial reference) note: vA= vCM + v⁰_A= 0, ∵ vA⁰ = ω× rCM →A=−v^CM IAα = M gR sin β , IA= ICM + M R² , aCM = αR

→ α, a^CM

Note: vA= 0, vB = 2vCM (refer to inertial reference)

Physical pendulum

rCM = 2 I0 = 1

12m ²+ m µ

2

¶2

= 1 3m ² Γ = rCM × mg = mg2sin θ· ¯ L = I0ω = I0

dθ

dt = I0ω· ⊗

θ

g mr

CM

O (pivod point)

2 l/

rr ωr ⊗

 equation of motion

I0

dω dt = I0

d²θ

dt² =−mg 2sin θ for reference: simple pendulum

m d²θ

dt² =−mg sin θ

small oscillation θ ¿ 1 sin≈ θ

∴ d²θ dt² =−

µmg 2I0

¶ θ

∴ θ (t) = θ^Acos (2πνt + δ)

→ ν = 1 2π

µmg 2I0

¶¹₂

= 1 2π

µ3g 2

¶¹₂

Gyro.

φ ω r

, ^d

φ

β O

refer.

axis

g Mr

CM CM

r L r r r

, , ω

φ ^ˆ Γ r ,

rr

LCM = ICMω0

Γ = rCM × Mg refer to O

Γ⊥ L^CM ∴ L^CM always changes direction only

→ L^CM rotates about a vertical axis with angular frequency ωp

if ωp ¿ ω⁰ L≈ L^CM

∵¯¯¯L^CM¯¯¯ = const.

∴ dLCM

dt = ωp× L^CM = rCM × Mg

→ ω^p =−MgrCM

LCM

precession

precession of e⁻ in a magnetic field B₀

B^r0^,ω^rL

µ^rs

e−

m,

Sr Γr

electron

• e⁻ ⇒ • m^e,−e, S, µ^s mass, charge, spin, magnetic moment S → angular momentum, quantized, µ^rs = ^N

S

permanent magnet; quantized both in magnitude & direction.

torque on electron Γ = µs× B⁰ , Γ ⊥ S

∴ dS

dt = µs× B⁰ ,

= ωL× S

µs =−γS , γ = e/m: gyro-magnetic ratio

→ ω^L = B0γ ωL: Larmor precessional frequency total kinetic energy

Ktotal =X

i

1

2miv_i² =X

i

1

2mi(vCM + v_i⁰)² = 1

2M v_CM² + 1 2

X

i

mi(v_i⁰)²

= KCM + Krot

Krot = 1 2

X

i

mi(v_i⁰)² =X

i

1

2mi(ωi× ri⁰)² = 1

2ICM · ω²

Coupled oscillation

κ

f r

O

vr

R

0 xr

= x

M

A

⊗ α,−→

OA× f^µ

¯ −→AO× (−κ · x)

approach 1. refer to O M aCM =−κx + f^µ ICM · α = Γ =−→OA× f^µ constraint condition

aCM = α· R ∵ ∆x = R∆θ , a^CM = d²x

dt² , α = d²θ dt²

∴ ma^CM =−κx − f^µ , ICM · α = f^µ· R ∴ f^µ= ICM

R · α = a^CMICM

R²

∴ aCM =− κ M + ICM

R

· x = − κ Mef f · x SHM

x = xAcos (2πvt + δ) , ν = 1 2π

µ κ Mef f

¶¹₂

approach 2. refer to A vA= 0

IA· α =−→AO× (−κx) IA= ICM + M R² = Mef f · R²

∴ IA· α = −R · κ · x IA

R² d²x

dt² =−κ · x ∴ ν = 1 2π

µ κ IA/R²

¶¹₂

approach 3. conservation of mechanical energy E = 1

2M v_CM² + 1

2ICMω²+1

2κx² = const

∵ ∆x = R · ∆θ , v^CM = dx

dt , ω = dθ dt

∴ v^CM = R· ω ∴ E = 1 2

µ

M +ICM

R²

¶

v_CM² + 1

2κx² = const

∵ dE

dt = 0 ∴ Mef f · a^CM + κ· x = 0 Collision between a particle & a rigid body

O

m vr

CM

, l M

b

θ

O

( ) ^θ

^{= 0}

ω

m

pivot

CM

conservation of L refer to O after collision m remains inside M

mv0· b = I⁰· ω (0) , I⁰ = 1

3M · ²+ m· b² rotational inertia w.r.t. O µ

M · 2+ m· b

¶

(1− cos θ^A)· g = 1

2I0· ω²(0)

Note: Force on point O 6= 0 , ∵point O is fixed constraint condition total momentum is not conserved.

Atwood Machine

1 1

,T a r r

M R

T r

2 2

g , a m r r g

m r

α ^r

⊗

m

m

tension of the string T1, T2

m1a1 = T1− m¹g m2a2 = m2g− T² ICM · α = (T²− T¹)· R

a1 = a2 = a

α· R = a , I^CM = 1 2M R² Note: T1 6= T²

→ a = (m2− m¹) g m1+ m2+ICM

R²