Angular momentum Vector product.

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1 Lecture 19 Chapter 11 Physics I Angular momentum Vector product. Course website: Lecture Capture:

2 Outline Chapter 11 Angular Momentum Vector Cross Product Conservation of Ang. Mom. Ang. Mom. of point particle Rigid Objects

3 Let s introduce Vector Cross Product

4 If we have two vectors Then the vector product is Vector Cross Product B A C A B Magnitude C A B ABsin Direction: perp. to both A and B (right hand rule) Order matters: C A B B A A B

5 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

6 ConcepTest 1 Vector product iˆ, ˆ, j kˆ For the unit vectors Find the following vector products 1) iˆ iˆ? 2) iˆ ˆj? C A B ABsin A) B) C) 1) iˆ iˆ 2) iˆ ˆj 1) iˆ iˆ 2) iˆ ˆj 1) iˆ iˆ 2) iˆ ˆj y iˆ 0 0 kˆ 0 ˆj iˆ iˆ iˆ ˆj 0 kˆ z x

7 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

8 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

9 Write Torque as the Cross Product F Let s look at a door top view: Axis of rotation r Applied force F produces torque rf sin Now, with vector product notation we can rewrite torque as r F Torque direction out of page (right hand rule) Notation convention: Direction out of the page Direction into the page

10 Angular Momentum

11 For translational motion we needed the concepts of force, F linear momentum, p mass, m For rotational motion we needed the concepts of torque, angular momentum, L moment of inertia, I Angular momentum is the rotational equivalent of linear momentum p mv L?

12 y Angular Momentum of a single particle z L r p x p O r r m p L r p Suppose we have a particle with -linear momentum -positioned at r O Then, by definition: Angular momentum of a particle about point O is L rpsin Carefull: Let s calculate angular momentum of m about point O L r p since r p, rpsin 0 so 0, sin 0 Thus, angular momentum of m O 0 L but L 0 O Cont.

13 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

14 By definition: L r p Example: projectile motion What is the angular momentum of a free particle of mass m moving near the surface of the Earth? y p 0 A O r p x Let s calculate the angular momentum of a particle about point O. 1) L at point O: L L r p 0 sin 0 L L r psin 0 2) L at point A: 0 So, the angular momentum is changing.

15 Example: the Earth around the Sun. What is the angular momentum of a particle of mass m moving with speed v=const in a circle of radius r in a counterclockwise direction? By definition: L r p L r p Let s calculate the angular momentum of a particle about point O. m p L θ=90 L L rpsin rmv So, the angular momentum is constant. The magnitude is O r const The direction of the angular momentum is perpendicular to the plane of circle (right-hand rule) But, again, L calculated relative to O is obviously not a constant. It depends on a choice of origin

16 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

17 Two definitions of Angular Momentum Single particle L r p L r p Rigid symmetrical body L I L

18 dl dl Rotational N. 2 nd law Let s find relationship between angular momentum and torque for a point particle: dr L r p p 0 mv r F v dl r dp N. 2 nd law F p mv Rotational N. 2nd law written in terms of L. Torque causes the particle s angular momentum to change dp

19 Rotational N. 2 nd law Let s show that this rotational N. 2 nd law is the same to the one presented in Lecture 18 I I d I We got exactly the same expression d ( I ) dl dl Translational N.2 nd law F F ma dp Rotational N.2 nd law I dl

20 Example:What is the angular momentum (about the origin) of an object of mass m dropped from rest.

21 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) 100,000( kˆ) 10,000( kˆ) 100,000 iˆ kgm kgm kgm s s s z x h y r v L r p mv( rsin )( kˆ) mvh( kˆ ) 100,000( k ˆ )

22 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

23 Thank you See you on Monday

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