-Angular Momentum of a Rigid Object-Angular Momentum of a Rigid Object-Conservation of Angular Momentum -Conservation of Angular Momentum

AP Physics CMrs. Coyle

Angular Momentum of a Particle on Angular Momentum of a Particle on Rotating Rigid ObjectRotating Rigid Object

• For a particle of mass mi of the object as shown

• Li = ri mi vi = miri2

• L and vectors are perpendicular to the xy plane.

Angular Momentum of a Rotating Angular Momentum of a Rotating Rigid ObjectRigid Object

• The angular momentum of the entire object, is the sum of the angular momentum vectors of all the individual particles

2z i i i

i i

L L m r

L I

Rotational Form of Newton’s Rotational Form of Newton’s Second LawSecond Law

extzdL dI

dt dt

ext I

Note:Note:

• The rotational form of Newton’s Second Law is also valid for a rigid object rotating about a moving axis provided the moving axis:(1) passes through the center of mass(2) is an axis of symmetry

Angular Momentum of a Bowling BallAngular Momentum of a Bowling Ball

• I=2/5MR 2

• Lz = I• The direction of

the angular momentum is in the positive z direction

Conservation of Angular Conservation of Angular MomentumMomentum

In the absence of a net external torque (the system is isolated) the net angular momentum of a system is conserved.

Li = Lf

Ii i = If f

Remember:

• Closed System: A system that has no gain nor loss of mass.• Isolated System: A closed system with

no net external force acting on it.

Ii i = If f • Why does

the skater spin faster when she pulls together her arms?

Video Link: Gyroscope

• http://www.youtube.com/watch?v=cquvA_IpEsA&NR=1

• Bicycle Wheel Gyroscope (M.I.T. Physics)• http://www.youtube.com/watch?v=8H98BgRz

pOM&NR=1

GyroscopeGyroscope

Ex: Conservation of Angular Momentum

Video Link: Conservation of Angular Momentum (Bicycle

Wheel)

• http://www.youtube.com/watch?v=UZlW1a63KZs&NR=1

Note:

For an isolated system: Energy,

Linear Momentum, and

Angular Momentum are conserved.

What happens to the angular speed of the system as

the person moves towards the center

of the Merry-Go-Round?

Σ Ii i = Σ If f

Why does the kinetic energy of the system increase when

the person moves

towards the center?

Motion of a TopPrecessional Motion : the motion of the axis of symmetry.

Angular Momentum of a Molecule

Ex: #30• A student sits on a freely rotating stool holding

two 3kg masses. When his arms are extended horizontally, the masses are 1m from the axis of rotation and he rotates with an angular speed of 0.750rad/s. The moment of inertia of the student plus stool is 3kg m2 and is assumed constant. The student pulls the weights inward horizontally to a position 0.3m from the rotation axis. a) Find the new angular speed of the student, b) The kinetic energy of the rotating system before and after he pulls the weights inwards. Ans: a) 1.91rad/s, b)2.53J , 6.44J