1. Physics 106: Mechanics Lecture 05 Wenda Cao NJIT Physics Department

2. February 18, 2011 Angular Momentum  Vectors  Cross Product  Torque using vectors  Angular Momentum

3. February 18, 2011 Vector Basics  We will be using vectors a lot in this course.  Remember that vectors have both magnitude and direction e.g.  You should know how to find the components of a vector from its magnitude and direction  You should know how to find a vector’s magnitude and direction from its components Ways of writing vector notation  y x axax ayay

4. February 18, 2011  y x z   Projection of a Vector in Three Dimensions  Any vector in three dimensions can be projected onto the x - y plane.  The vector projection then makes an angle  from the x axis.  Now project the vector onto the z axis, along the direction of the earlier projection.  The original vector a makes an angle  from the z axis. y x z 

5. February 18, 2011 Vector Basics  You should know how to generalize the case of a 2-d vector to three dimensions, e.g. 1 magnitude and 2 directions  Conversion to x, y, z components  Conversion from x, y, z components  Unit vector notation:  y x z 

6. February 18, 2011 A Note About Right-Hand Coordinate Systems  A three-dimensional coordinate system MUST obey the right-hand rule.  Curl the fingers of your RIGHT HAND so they go from x to y. Your thumb will point in the z direction. y x z

7. February 18, 2011 Vector Math  Vector Inverse Just switch direction  Vector Addition Use head-tail method, or parallelogram method  Vector Subtraction Use inverse, then add  Vector Multiplication Three kinds! Multiplying a vector by a scalar Scalar, or dot product Vector, or cross product Vector Math by Components

8. February 18, 2011 Scalar Product of Two Vectors  The scalar product of two vectors is written as It is also called the dot product   is the angle between A and B  Applied to work, this means

9. February 18, 2011 Dot Product  The dot product says something about how parallel two vectors are.  The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other.  Components 

10. February 18, 2011 Projection of a Vector: Dot Product  The dot product says something about how parallel two vectors are.  The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other.  Components  Projection is zero

11. February 18, 2011 Derivation  How do we show that ?  Start with  Then  But  So

12. February 18, 2011 Cross Product  The cross product of two vectors says something about how perpendicular they are.  Magnitude:  is smaller angle between the vectors Cross product of any parallel vectors = zero Cross product is maximum for perpendicular vectors Cross products of Cartesian unit vectors:  y x z i j k i kj

13. February 18, 2011 Cross Product  Direction: C perpendicular to both A and B (right-hand rule) Place A and B tail to tail Right hand, not left hand Four fingers are pointed along the first vector A “sweep” from first vector A into second vector B through the smaller angle between them Your outstretched thumb points the direction of C  First practice

14. February 18, 2011 More about Cross Product  The quantity ABsin  is the area of the parallelogram formed by A and B  The direction of C is perpendicular to the plane formed by A and B  Cross product is not commutative  The distributive law  The derivative of cross product obeys the chain rule  Calculate cross product

15. February 18, 2011 Derivation  How do we show that ?  Start with  Then  But  So

16. February 18, 2011  The torque is the cross product of a force vector with the position vector to its point of application  The torque vector is perpendicular to the plane formed by the position vector and the force vector (e.g., imagine drawing them tail-to-tail)  Right Hand Rule: curl fingers from r to F, thumb points along torque. Torque as a Cross Product Superposition :  Can have multiple forces applied at multiple points.  Direction of  net is angular acceleration axis

17. February 18, 2011 Calculating Cross Products Solution: i kj Calculate torque given a force and its location Solution: Find: Where:

18. February 18, 2011 i kj Net torque example: multiple forces at a single point x y z  3 forces applied at point r : Find the net torque about the origin: Here all forces were applied at the same point. For forces applied at different points, first calculate the individual torques, then add them as vectors, i.e., use: oblique rotation axis through origin

19. February 18, 2011 Angular Momentum  Same basic techniques that were used in linear motion can be applied to rotational motion. F becomes  m becomes I a becomes  v becomes ω x becomes θ  Linear momentum defined as  What if mass of center of object is not moving, but it is rotating?  Angular momentum

20. February 18, 2011 Angular Momentum I  Angular momentum of a rotating rigid object L has the same direction as  L is positive when object rotates in CCW L is negative when object rotates in CW  Angular momentum SI unit: kgm 2 /s  Calculate L of a 10 kg disc when  = 320 rad/s, R = 9 cm = 0.09 m  L = I  and I = MR 2 /2 for disc  L = 1/2MR 2  = ½(10)(0.09) 2 (320) = 12.96 kgm 2 /s

21. February 18, 2011 Angular Momentum II  Angular momentum of a particle r is the particle’s instantaneous position vector p is its instantaneous linear momentum Only tangential momentum component contribute r and p tail to tail form a plane, L is perpendicular to this plane

22. February 18, 2011 Angular Momentum of a Particle in Uniform Circular Motion  The angular momentum vector points out of the diagram  The magnitude is L = rp sin  = mvr sin (90 o ) = mvr  A particle in uniform circular motion has a constant angular momentum about an axis through the center of its path O Example: A particle moves in the xy plane in a circular path of radius r. Find the magnitude and direction of its angular momentum relative to an axis through O when its velocity is v.

23. February 18, 2011 Angular momentum III  Angular momentum of a system of particles angular momenta add as vectors be careful of sign of each angular momentum for this case:

24. February 18, 2011 Example: calculating angular momentum for particles Two objects are moving as shown in the figure. What is their total angular momentum about point O? m2m2 m1m1

25. February 18, 2011 Angular Momentum and Torque  Net torque acting on an object is equal to the time rate of change of the object’s angular momentum  Angular momentum is defined as  Analog in impulse

26. February 18, 2011 SUMMARY Translation Force Linear Momentum Kinetic Energy Rotation Torque Angular Momentum Kinetic Energy Linear Momentum Second Law Angular Momentum for rigid bodies about common fixed axis Second Law Momentum conservation - for closed, isolated systems Systems and Rigid Bodies Apply separately to x, y, z axes