LE 0110b

1
The Right-Hand Rule

• convert a sense of swirl to a tipped arrow by
  letting right hands fingers close in the swirl
  sense thumb points along vector
• here, applied to angular velocity w to
  vectorize it to w

wf
y
wi
x
2
Some instances of the VECTOR version of a dw/dt
3
Vectorizing the torque t is along the axis of
twist, so its normal to the plane of F and r (a
unique direction), with tip given by

• The dreaded RIGHT HAND RULE
• torque t r F contains both direction and
  magnitude

4
The right-hand rule and the vector cross-product

• given vectors A and B
• tail-to-tail angle q (smaller)
• C A B
• magnitude C AB sin q
• direction swing first (A) into second (B)
  through q, letting fingers of right hand close in
  that sense thumb points along C (normal to plane
  of A, B)
• you may have to contort!!
• direction a normal peanut butter jar lid moves
  as you tighten it

5
An alternative formula for the cross-product

• cool facts A A 0 A B - (B A)
• (anti)parallel vectors have A B 0
• perpendicular vectors have A B AB
• compare to dot product AB AB cos q

6
Angular Momentum

• rotational analog to translational momentum p
• start with angular momentum for one point mass
• lets see . . . since F dp/dt and since t r
  F
• so t r F r (dp/dt) d(r p)/dt if
  moment arm doesnt change
• define angular momentum L r p
• N2 for angular motion t dL/dt
• for fixed axis t Ia so since a dw/dt, we see
  that L Iw (nice analogous to p mv)
• in many cases (but by no means all cases) L I
  w
• if this is the situation, then L w
• we will see that t dL/dt (rotational version
  of N2)

7
A mass moves in U.C.M. Put origin at center (it
depends!!)
L r p
Despite the changing directions of r and p, L is
a CONSTANT here
8
The Behavior of L for a System of Point Masses

• In first term, vi dri/dt so term is zero
• In second term, ai dvi/dt Fi/mi where
• Fi sum of external internal forces on body i

9
Sorting out the Torques for a System of Point
Masses

• first term is just sum of external torques
• second term is zero, as we now show

N3 F i,j - F j,i so we have pairs of terms
So if sum of external torques is zero, and if N3
applies, and if the lines of the internal forces
are along the line joining particles, then
Angular Momentum is Conserved!
10
System Angular Momentum Conservation

• bodys (or systems) L is a vector that points
  along w system momentum is conserved NO MATTER
  WHAT KIND OF INTERNAL FORCES ACT!! INCLUDING
  NON-CONSERVATIVE FORCES
• more universal than (system) energy conservation
• applies to planets in their orbits (Keplers
  equal-areas law)
• applies to spinning systems skaters, gas
  clouds, superfluids

11
L is Conserved
12
Chapter 11 43

• 8 60-kg skaters in a line, moving at 4.6 m/s
• Assume they form a long thin rod!!

12 m

• Find L about the origin L Mvr
• L is conserved
• now L I w ? solve for w