1. Physics 215 – Fall 2014Lecture 06-21 Welcome back to Physics 215 Today’s agenda: Friction, drag Tension, pulleys Forces in circular motion Impulse

2. Physics 215 – Fall 2014Lecture 06-22 Current homework assignment HW5: –Knight textbook Ch.6: 38, 42, 56, 58 –Ch.7: 46, 54 –due Friday, Oct. 3 rd in recitation

3. Physics 215 – Fall 2014Lecture 06-23 Summary of friction 2 laws of friction: static and kinetic Static friction tends to oppose motion and is governed by inequality F s ≤  s N Kinetic friction is given by equality F K =  K N

4. Physics 215 – Fall 2014Lecture 06-24 The magnitude of the force of kinetic friction between two objects depends on the type of surfaces of the objects depends on the normal force that the objects exert on each other does not depend on the surface area where the two objects are touching does not depend on the speed with which one object is moving relative to the other What if  > tan -1  s ?

5. Physics 215 – Fall 2014Lecture 06-25 What if  > tan -1  s ? Block begins to slide Resolve along plane: Wsin  -   Wcos  = ma Or: a = g(sin  -   cos  )

6. Physics 215 – Fall 2014Lecture 06-26 Tension For an ideal string or rope connecting two objects: does not stretch  inextensible has zero mass Let’s look at an example of a cart connected to a falling mass by an ideal string...

7. Physics 215 – Fall 2014Lecture 06-27 1.equal to 1.0 N (i.e. the weight of A), 2.between 1.0 N and 1.5 N, 3.equal to 1.5 N (i.e. the weight of C), or 4.equal to 2.5 N (i.e. the sum of their weights)? Blocks A and C are initially held in place as shown. After the blocks are released, block A will accelerate up and block C will accelerate down. The magnitudes of their accelerations are the same. Will the tension in the string be

8. Physics 215 – Fall 2014Lecture 06-28 Demo: Pulleys F W T *2 pulleys 2T = W F = T = W/2 *N pulleys F = W/N!

9. Physics 215 – Fall 2014Lecture 06-29 Forces in circular motion Motion around circular track, constant speed (for now): a rad = v 2 /r

10. Physics 215 – Fall 2014Lecture 06-210 1.four times as great 2.twice as great 3.equal to 4.one half as great as the acceleration of ball A. Two identical balls are connected by a string and whirled around in circles of radius r and 2r at angular speed. The acceleration of ball B is

11. Physics 215 – Fall 2014Lecture 06-211 1.less than 2.equal to 3.greater than the tension in string R. The two identical balls are whirled around in a circle as before. Assume that the balls are moving very fast and that the two strings are massless. The tension in string P is

12. Physics 215 – Fall 2014Lecture 06-212 Car on circular track with friction Motion around circular track, constant speed (for now): a rad = v 2 /r

13. Physics 215 – Fall 2014Lecture 06-213 Motion of car on banked circular track (no friction) R Speed v  car N W Horizontal forces: Vertical: a =

14. Physics 215 – Fall 2014Lecture 06-214 Motion on loop-the-loop car What is normal force on car at top and bottom of loop? Neglect friction; assume moves with speed v B at bottom and v T at top At bottomAt top

15. Physics 215 – Fall 2014Lecture 06-215 Impulse Constant force F 12 acting on object 1 due to object 2 for a time  t yields an impulse I 12 = F 12  t In general, for a time varying force need to use this for small  t and add: I =  F(t)  t =

16. Physics 215 – Fall 2014Lecture 06-216 Impulse for time varying forces * area under curve equals impulse t F(t) t1t1 t2t2

17. Physics 215 – Fall 2014Lecture 06-217 Impulse  change in momentum Consider first constant forces... Constant acceleration equation: v f = v i + at mv f - mv i = mat = If we call p = mv momentum we see that  p =

18. Physics 215 – Fall 2014Lecture 06-218 Impulse demo Cart equipped with force probe collides with rubber tube Measure force vs. time and momentum vs. time Find that integral of force curve is precisely the change in p!

19. Physics 215 – Fall 2014Lecture 06-219 Momentum of an object: p = mv Definitions of impulse and momentum Impulse imparted to object 1 by object 2: I 12 = F 12  t

20. Physics 215 – Fall 2014Lecture 06-220 The net impulse imparted to an object is equal to its change in its momentum. Impulse-momentum theorem I net =  p

21. Physics 215 – Fall 2014Lecture 06-221 1.Case A. 2.Case B. 3.Case C. 4.Cases A and B. Consider the change in momentum in these three cases: A.A ball moving with speed v is brought to rest. B.The same ball is projected from rest so that it moves with speed v. C.The same ball moving with speed v is brought to rest and immediately projected backward with speed v. In which case(s) does the ball undergo the largest change in momentum?

22. Physics 215 – Fall 2014Lecture 06-222 Reading assignment Momentum, collisions Chapter 9 in textbook