Lecture 19 Chapter 11 Physics I 11.20.2013 Angular momentum Vector product. Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html
Outline Chapter 11 Angular Momentum Vector Cross Product Conservation of Ang. Mom. Ang. Mom. of point particle Rigid Objects
If we have two vectors Then the vector product is Magnitude Vector Cross Product A A x î A y ĵ A z ˆk B B x î B C A y ĵ B B z ˆk C A B ABsin Direction: perp. to both A and B (right hand rule) B C B A A Order matters: A B B A
θ=0 A A B 0 B The vector product is zero when vectors are parallel Cross product C A B ABsin A 1 B 2 AB B A 1 B 2 θ=30 A AB The vector product increases C A B A B B AB θ=90 A A B AB The vector product is max when vectors are perpendicular The cross product vector increases from 0 to AB as θ increases from 0 to 90
ConcepTest 1 Vector product, ˆ, j kˆ For the unit vectors Find the following vector products 1)? 2) ˆj? C A B ABsin A) B) C) 1) 2) ˆj 1) 2) ˆj 1) 2) ˆj y 0 0 kˆ 0 ˆj ˆj 0 kˆ z x
A A x î A y ĵ A z ˆk Vector Cross Product ˆ ˆ ˆ ˆ i i i i Sin0 0 ˆ ˆ ˆ ˆ i j i j Sin90 1 î î 0 ĵ ĵ 0 ˆk ˆk 0 î ĵ ˆk ĵ ˆk î ˆk î ĵ B B x î B y ĵ B z ˆk A B (A x î A y ĵ A z ˆk) (Bx î B y ĵ B z ˆk) A x B x (î î ) A B (î ĵ) A B x y x z (î ˆk) A y B x ( ĵ î ) A yb y ( ĵ ĵ) A yb z ( ĵ ˆk) A z B x ( ˆk î ) A zb y ( ˆk ĵ) A zb z ( ˆk ˆk) (A y B z A z B y )î (A zb x A x B z ) ĵ (A xb y A y B x ) ˆk
Vector Cross Product. Example What is the vector cross product of the two vectors: A 1î 2 ĵ 4 ˆk B 2î 3ĵ 1ˆk A B (A y B z A z B y )î (A zb x A x B z ) ĵ (A xb y A y B x ) ˆk A B [(2 1) (4 3)]î [(4 2) (11)] ĵ [(13) (2 2)]ˆk A B 14î 9 ĵ 1 ˆk
Write Torque as the Cross Product Let s look at a door top view: r F sin F Axis of rotation Applied force F produces torque Now, with vector product notation we can rewrite torque as rf rf sin r F Torque direction out of page (right hand rule) Notation convention: Direction out of the page Direction into the page
Angular Momentum Angular momentum is the rotational equivalent of linear momentum p mv L?
z Angular Momentum of a single particle y O r m L r p p L r p i i x Suppose we have a particle with -linear momentum -positioned at r Then, by definition: Angular momentum of a particle about point O is L r m v If we have many particles, the total angular momentum is L L L 1 L2 L3... p
Angular Momentum of a single particle. Example What is the angular momentum of a particle of mass m moving with speed v in a circle of radius r in a counterclockwise direction? By definition: Angular momentum of a particle about point O is L r p L r p Let s rewrite this result slightly. v r L L Recall: L rmv mr( r) L p r θ=90 rpsin rmv 2 ( mr ) where 2 I mr L I It looks like we can get a different expression for L Next
Angular Momentum of a rigid body For the rotation of a symmetrical object about the symmetry axis, the angular momentum and the angular velocity are related by (without a proof) L I L I I moment of inertia of a body L points towards L I L I
Two definitions of Angular Momentum Single particle L r p L r p Rigid symmetrical body L I L
Angular Momentum and Torque/particles Let s find relationship between angular momentum and torque for a point particle: dl dl L r p dr dp p r 0 v mv r F N. 2 nd law F p m v dl dp Torque causes the particle s angular momentum to change dl d i L i i For many particles: net dl net
Angular Momentum and Torque/rigid body Let s find the same relationship between angular momentum and torque for a rigid body: net We got exactly the same expression I I I net d d ( I ) dl dl net Torque causes angular momentum to change
Example:What is the angular momentum (about the origin) of an object of mass m dropped from rest.
Example: What is the torque (about the origin) of an object of mass m dropped from rest.
ConcepTest 2 traffic light/car A car of mass 1000 kg drives away from a traffic light h=10 m high, as shown below, at a constant speed of v=10 m/s. What is the angular momentum of the car with respect to the light? A) B) C) 10,000( kˆ) 10,000( kˆ) 100,000 kgm kgm kgm 2 2 2 s s s z x h y r v L r p mv( rsin )( kˆ) mvh( kˆ) 10,000( kˆ)
Conservation of Angular Momentum Angular momentum is an important concept because, under certain conditions, it is conserved. dl If the net external torque on an object is zero, then the total angular momentum is conserved. If net net dl 0, then L const 0 L I L 1 L For a rigid body 2 I 1 1 I 2 2
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