2 IN THIS CHAPTER, you will continue discussing rotational dynamics Today we are going to discuss: Chapter 12: Cross Product/Torque: Section Angular Momentum: Section 12.11
3 Torque (Refreshing) r F It should depend on force, lever arm, angle: rf sin Torque is a turning force (the rotational equivalent of force). Axis of rotation Sign of a torque convention: A positive torque tries to rotate an object in a CCW direction A negative torque tries to rotate an object in a CW direction
4 Example Torque The body shown in the figure is pivoted at O. Three forces act on it in the direction shown in the figure: FA=10 N at point A, 8.0 m from O; FB=10 N at point B, 4.0 m from O and FC=19 N at point C, 3.0 m from O. Calculate the net torque about point O.
5 Newton s 2 nd law of rotation Force causes linear acceleration: (Translational N.2 nd law) Torque causes angular acceleration: (Rotational N.2 nd law) F ma I Torque (rotational equivalent of force) Angular acceleration I is the Moment of Inertia (rotational equivalent of mass)
6 Example Pulley and mass An 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?
7 Notation for Vectors Perpendicular to the Page Physics requires a three-dimensional perspective, but two-dimensional figures are easier to draw. We will use the following notation:
8 rf sin This construction is quite common in Physics, so mathematicians decided to abbreviate it and give a nice name - Vector Cross Product
9 The Cross Product From now on, some equations will have a cross product =(ab sin θ, direction given by the right-hand rule) [I] Point fingers in the direction of the 1 st (a) vector, then bend them in the direction of the 2 nd one (b). The outstretched thumb will give a direction of the cross product [II] Use three fingers as shown in the figure The cross product of vectors a and b is a vector perpendicular to both a and b. Order is important in the cross product:
10 Cross product C A B ABsin C A B θ=0 A A B 0 B The vector product is zero when vectors are parallel A 1 B 2 θ=30 A AB B The vector product increases B 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
11 ConcepTest Cross Product Find a direction of a cross product
12 Now we can write Torque as a vector product F r rf sin Torque is a turning force (the rotational equivalent of force). Axis of rotation Now, with the vector product r notation F we can rewrite torque as Torque direction out of page (right hand rule)
13 Angular Momentum We will introduce angular momentum of A point mass m A rigid object
14 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?
15 y O O Angular Momentum of a single particle L r Suppose we have a particle with p -linear momentum -positioned at r x p z r r L m p Then, by definition: Angular momentum of a particle about point O is L r p L rpsin Carefull: Let s calculate angular momentum of m about point O r p since r p, rpsin 0 so 0, sin 0 Thus, angular momentum of m LO 0 but L O 0 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
16 Example Angular momentum (about the origin) of an object of mass m dropped from rest.
17 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
18 Two definitions of Angular Momentum Single particle L r p L r p Rigid symmetrical body L I L
19 Rotational N. 2 nd law Let s rewrite our rotational Newton s 2 nd Law in terms of angular momentum: I I d dt d ( I ) dt dl dt dl dt (We use the angular momentum expression for a rigid body but it can also be shown for a point mass) dl dt Rotational N. 2nd law written in terms of L. Torque causes the particle s angular momentum to change
20 Translational vs- Rotational N. 2 nd law Translational N.2 nd law F F ma dp dt Rotational N.2 nd law I dl dt