lecture 21 ch12 f19 angular momentum · let’s find relationship between angular momentum and...
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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/
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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?!
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
Angular velocity as a vector
Themagnitudeoftheangularvelocityvectorisω.
Theangularvelocityvector pointsalongtheaxisofrotationinthedirectiongivenbytheright‐handrule(RHR#2)asillustrated.
Amoregeneraldescriptionofrotationalmotionrequiresustoreplacethescalarsωandτwiththevectorquantities and
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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)
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
Angular Momentum of a rigid body
L I
points towardsL
Fortherotationofasymmetricalobjectaboutthesymmetryaxis,theangularmomentumandtheangularvelocityarerelatedby(withoutaproof)
IL
IL
IL
I – moment of inertia of a body
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
L
Two “definitions” of Angular Momentum
r
p
L
L I
L
r p
Rigid symmetrical bodySingle particle
Summary
End of Class
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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)
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
Translational – vs- Rotational N. 2nd law
amF
dtpdF
I
Translational N.2nd law Rotational N.2nd law
dtLd
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
Angular momentum (about the origin) of an object of mass m dropped from rest (cont.).
Example
(cont.)
DepartmentofPhysicsandAppliedPhysicsPHYS.1410Lecture21A.Danylov
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