Cross Product Angular Momentum -...

Department of Physics and Applied PhysicsPHYS.1410 Lecture 21 Danylov

Lecture 21

Chapter 12

Cross ProductAngular Momentum

Physics I

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI


Today we are going to discuss:

Chapter 12:

Cross Product/Torque: Section 12.10 Angular Momentum: Section 12.11

IN THIS CHAPTER, you will continue discussing rotational dynamics


Torque is a turning force (the rotational equivalent of force).

It should depend on force, lever arm, angle:

(Refreshing)

rF sinF

Torque

Axis of rotation

Sign of a torque convention:A positive torque tries to rotate an object in a CCW directionA negative torque tries to rotate an object in a CW direction

r


TorqueExampleThe body shown in the figure is pivoted at O. Three forces act on it in the direction shown in the figure: FA=10 N at point A, 8.0 m from O; FB=10 N at point B, 4.0 m from O and FC=19 N at point C, 3.0 m from O. Calculate the net torque about point O.


Torque causes angular acceleration:

Force causes linear acceleration: (Translational N.2nd law)

Newton’s 2nd law of rotation

I

I is the Moment of Inertia(rotational equivalent of mass)

amF

Angular accelerationTorque

(rotational equivalent of force)

(Rotational N.2nd law)


Pulley and massExample

An object of mass m is hung from a cylindrical pulley of radius R and mass M and released from rest. What is the acceleration of the object?


Notation for Vectors Perpendicular to the Page

Physics requires a three-dimensional perspective, but two-dimensional figures are easier to draw. We will use the following notation:


Vector Cross Product

rF sinThis construction is quite common in Physics, so mathematicians decided to abbreviate it and give a nice name -


The Cross Product

=(ab sin θ, direction given by the right-hand rule)

From now on, some equations will have a cross product

The cross product of vectors a and b is a vector perpendicular to both a and b.

[I] Point fingers in the direction of the 1st (a) vector, then bend them in the direction of the 2nd one (b).The outstretched thumb will give a direction of the cross product

[II] Use three fingers as shown in the figure

Order is important in the cross product:


Cross product

The cross product vector increases from 0 to AB as θ increases from 0 to 90

A

B

C

A

B

C A

B ABsin

0BA

θ=30

θ=0

A

B

θ=90A

B

The vector product is zero when vectors are parallel

The vector product increases The vector product is max when vectors are perpendicular

ABBA

ABBA 21

ConcepTest Cross Product

Find a direction of a cross product


Torque is a turning force (the rotational equivalent of force).

rF sin

F

Now we can write Torque as a vector product

Axis of rotation

r

r

F

Now, with the vector product notation we can rewrite torque as

Torque direction – out of page (right hand rule)


Angular MomentumWe will introduce angular momentum of

• A point mass m

• A rigid object


Angular momentum is the rotational equivalent of linear momentum

?L

vmp

For translational motion we needed the concepts of

force, Flinear momentum, p

mass, m

For rotational motion we needed the concepts of

torque, angular momentum, L

moment of inertia, I


Angular Momentum of a single particle

L

r p rpSinL

xz

y

O

r pm

L

r p Suppose we have a particle with-linear momentum -positioned at r

p

Then, by definition: Angular momentum of a particle about point O is

OCarefull: Let’s calculate angular momentum of m about point O’

prL sinpr

r

,since pr

0

0sin,0 so

Thus, angular momentum of m 0OL but 0OLAngular Momentum is not an intrinsic property of a particle.

It depends on a choice of origin

So, never forget to indicate which origin is being used


Angular momentum (about the origin) of an object of mass m dropped from rest.

Example


Angular Momentum of a rigid body

L I

points towardsL

For the rotation of a symmetrical object about the symmetry axis, the angular momentum and the angular velocity are related by (without a proof)

IL

IL

IL

I – moment of inertia of a body


Two definitions of Angular Momentum

L

r

p

L

L I

L

r p

Rigid symmetrical bodySingle particle


Rotational N. 2nd lawLet’s rewrite our rotational Newton’s 2nd Law in terms of angular momentum:

dtLd

Torque causes the particle’s angular momentum to change

Rotational N. 2nd lawwritten in terms of L.

dtLd

IdtdI

dtId )(

dtLd

(We use the angular momentum expression for a rigid body but it can also be shown for a point mass)


Translational – vs- Rotational N. 2nd law

amF

dtpdF

I

Translational N.2nd law Rotational N.2nd law

dtLd


Thank youSee you on Wednesday

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