lecture 21 ch12 f19 angular momentum · let’s find relationship between angular momentum and...

DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov

Lecture 21

Chapter 12

Angular Momentum

Physics I

Another Law? Am I in a Law school?

Course website:https://sites.uml.edu/andriy-danylov/teaching/physics-i/


Today we are going to discuss:

Chapter 12:

Angular Momentum: Section 12.11 Rotational Newton’s 2nd Law (general form): Section 12.11

IN THIS CHAPTER, you will continue discussing rotational dynamics


Torque due to Gravity

gMW CMR

The proofCM

We often encounter systems in which there is a torque exerted by gravity.

(Read only if you want)

The torque due to gravity is found by treating the object as if all its mass is concentrated at the center of mass.

An object will balance on a pivot only If the CM is directly above the pivot point. If the pivot point is not under the CM, the grav. torquewill cause the object to rotate

Example

MgSinRCMgrav


Torque causes angular acceleration:

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

Newton’s2nd lawofrotation

I

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

amF

Angular accelerationTorque

(rotational equivalent of force)

(Rotational N.2nd law)

Newton 2nd Law again?!


Pulley and massExampleAn 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?

ConcepTest A puckA) Steadilyincreases

B) Increasesforawhile,thenholdssteady

C)Holdssteady

D)Decreasesforawhile,thenholdssteady

Astudentgivesaquickpushtoapuckthatcanrotateinahorizontalcircleonafrictionlesstable.Afterthepushhasended,thepuck’sangularspeed(ω)

Atorquecausesangularaccelerationwhichleadstochangesoftheangularvelocity.Withnotorque,theangularvelocitystaysthesame.

221 IKrot


Angular velocity as a vector

Themagnitudeoftheangularvelocityvectorisω.

Theangularvelocityvector pointsalongtheaxisofrotationinthedirectiongivenbytheright‐handrule(RHR#2)asillustrated.

Amoregeneraldescriptionofrotationalmotionrequiresustoreplacethescalarsωandτwiththevectorquantities and


Angular momentum is the rotational equivalent of linear momentum

?L

vmp

For translational motion we needed the concepts of

For rotational motion we needed the concepts of

We will introduce angular momentum of • A point mass m • A rigid object

NowweneedtointroduceanotheroneofthemostimportantquantitiesinPhysics

Angular Momentum

force, F

linear momentum mass, m

torque,

angular momentum,moment of inertia, I


Angular Momentum of a single particle

L

r prpSinL

x

z

yO

r pm

L

r pSupposewehaveaparticlewith

massmand

linearmomentump

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

Angular 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

Thelargevalueoftheangularmomentumforamovingobject,themoreeasilytheobjectcansetanotherobjectinrotationalmotion.

Magnitude:Direction: Right‐handrule(RHR#1)


Angular Momentum of a rigid body

L I

points towardsL

Fortherotationofasymmetricalobjectaboutthesymmetryaxis,theangularmomentumandtheangularvelocityarerelatedby(withoutaproof)

IL

IL

IL

I – moment of inertia of a body


L

Two “definitions” of Angular Momentum

r

p

L

L I

L

r p

Rigid symmetrical bodySingle particle

Summary

End of Class


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

Example

(The shortest distance between the origin and the line of motion)

ConcepTest Traffic light/carA car of mass 10 kg drives away from atraffic light h=10 m high, as shown below, ata constant speed of v=10 m/s. What is theangular momentum of the car with respectto the traffic light?

A) B) C)

skgmk 2 )ˆ(1000

x

y

z

mh 10

skgmi 2 ˆ1000

smv /10

prL

skgmk 2 )ˆ(1000

r

)ˆ)(( krSinmv )ˆ( kmvh )ˆ(1000 k

kgm 10


Rotational N. 2nd law

Let’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.

I

dtdI

dtId )(

dtLd

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


Translational – vs- Rotational N. 2nd law

amF

dtpdF

I

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

dtLd


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

Example

(cont.)


Thank you


Rotational N. 2nd law

L

r p dtLd

dtLd

Let’s find relationship between angular momentum and torque for a point particle:

dtpdFlawndN

2.

vmp

dtLd

Torque causes the particle’s angular momentum to change

Rotational N. 2nd lawwritten in terms of L.

p

dtrd

dtpdr

vmv Fr

Read if only if you want

lecture 21 ch12 f19 angular momentum · let’s find relationship between angular momentum and...

Documents