Angular Momentum in Physics 211 – Spring 2003, Study notes of Physics

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Welcome back to Physics 211

• Angular velocity and acceleration as vectors
• Vector Product
• Torque as a vector
• Angular Momentum
• Conservation of Angular Momentum

Angular Velocity as a Vector Line of ω − gives rotational axis Arrow of ω − distinguishes between clockwise and counter-clockwise rotation

ω

ω

ω

ω

right hand thumb rule

right-handed screw rule

Rotating object

Arrow of ω given by:

Torque as a Vector?

Ι α = τ

α is a vector ⇒^ τ is a vector

Physics 211 – Spring 2003 Angular Momentum 7

Scalar and Vector Product of

Two Vectors

θ

b

a

Scalar Product (^) Vector Product

a b = c⋅

r r (“dot” product)^ (“cross” product)

c = a b cos?

result is a number result is a vector

a × b = c

r r r

c = a b sin?

θ

b

a

c

Use right hand thumb rule or right-handed screw rule to get the arrow of c

Magnitude: Direction: Along the line perpendicular to a and b

Torque as a Vector

point P

F

rP

t P = rP × F

r^ r^ r

Torque with respect to point P

t P = rP F sin θ

θ

Torque components

point P

F t^ P^ =^ rP^ × F

r^ r^ r

rP¦

τ¦ = r- x F

F

r-

For force with no component along the axis of rotation

rP

F

τ (^) P - = rP¦ x F

Axis of rotation

t (^) P = (rP (^) P + r⊥ )×F

r r r r

t P = rP P × F + r⊥ ×F

r r^ r^ r r

t P = t P⊥ + tP

r r r

Torque component along the axis of rotation changes magnitude of angular velocity “Torque about the axis of rotation”

Torque component perpendicular to the axis of rotation changes direction of angular velocity i.e. tries to tilt the axis of rotation about point P

Two Simple Motions With Constant

Angular Momentum

• Linear Motion with constant velocity

P

p

p p = m v (^) constant

L = d p constant

d

• Circular Motion with constant speed

P

p R

p

R

p changes direction

L = R p constant

Newton’s 2nd^ Law in Angular Form

dL d(r p)

dt dt

×

r (^) r r

dr dp p + r dt dt

= × ×

r r r r

= v × m v + r × F

r r r r

= 0 + t

r

dL t dt

=

r r

Conservation of Angular

Momentum

L tot Li

i

= (^) ∑

r r

tot external net

dL

t

dt

r

r

P tot pi

i

= (^) ∑

r r

tot external net

dP

F

dt

r

r

Angular Momentum (^) Linear Momentum

t external net = 0

r

L tot = const

r

⇒

If there is no net external torque on the system, total angular momentum of the system is conserved

L tot i = Ltot f

r r