Welcome back to Physics 215

1. Welcome back to Physics 215 Today’s agenda: Friction, drag Tension, pulleys Forces in circular motion Impulse

2. Current homework assignment • HW5: • Knight textbook Ch.6: 38, 42, 56, 58 • Ch.7: 46, 54 • due Friday, Oct. 3rd in recitation

3. Summary of friction • 2 laws of friction: static and kinetic • Static friction tends to oppose motion and is governed by inequality Fs ≤ msN • Kinetic friction is given by equality FK = mKN

4. What if  > tan-1ms ? The magnitude of the forceofkinetic 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

5. What if  > tan-1ms ? • Block begins to slide • Resolve along plane: Wsinq- mKWcosq= ma • Or: a = g(sinq- mKcosq)

6. For an ideal string or rope connecting two objects: • does not stretch  inextensible • has zero mass Tension • Let’s look at an example of a cart connected to a falling mass by an ideal string...

7. 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 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)?

8. Demo: Pulleys *2 pulleys 2T = W F = T = W/2 T F *N pulleys F = W/N! W

9. Motion around circular track, constant speed (for now): Forces in circular motion arad = v2/r

10. 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 1. four times as great 2. twice as great 3. equal to 4. one half as great as the acceleration of ball A.

11. 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 1. less than 2. equal to 3. greater than the tension in string R.

12. Motion around circular track, constant speed (for now): Car on circular track with friction arad = v2/r

13. Motion of car on banked circular track (no friction) car R N a = q W Speed v Horizontal forces: Vertical:

14. Motion on loop-the-loop What is normal force on car at top and bottom of loop? Neglect friction; assume moves with speed vB at bottom and vT at top car At bottom At top

15. Impulse • Constant force F12 acting on object 1 due to object 2 for a time Dt yields an impulse I12 = F12t • In general, for a time varying force need to use this for small Dt and add: I =  F(t)t =

16. Impulse for time varying forces F(t) * area under curve equals impulse t t2 t1

17. Impulse  change in momentum • Consider first constant forces ... • Constant acceleration equation: vf = vi + at mvf - mvi = mat = • If we call p = mv momentum we see that p =

18. 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. Definitions of impulse and momentum Impulse imparted to object 1 by object 2: I12 = F12t Momentum of an object: p = mv

20. Impulse-momentum theorem Inet = p The net impulse imparted to an object is equal to its change in its momentum.

21. 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. 1. Case A. 2. Case B. 3. Case C. 4. Cases A and B. In which case(s) does the ball undergo the largest change in momentum?

22. Reading assignment • Momentum, collisions • Chapter 9 in textbook