1. Lecture 24 ENGR-1100 Introduction to Engineering Analysis

2. Today’s Lecture Outline Dry friction (Coulomb friction)

3. Friction Friction forces oppose the tendency of contacting surfaces to slip one relative to the other. Dry friction- the tangential component of the contact force.

4. Friction Forces have to be concurrent in order to be in equilibrium.

5. Friction forces as a function of P Maximum value of friction forces is called the limiting value of static friction. This condition is also called the impending motion.

6. Tip vs. slip Body Tips F1F1 P2>P1P2>P1 F2F2

7. The magnitude of the friction forces F max =  s N Where  s is the coefficient of static friction. It is independent of normal forces and area of contact. F<  s N The general case for equilibrium conditionOnce the body starts to slip F=  k N  k is the coefficient of dynamic friction  k <  s

8. The Resultant of the friction and normal forces R= N 2 +F 2 tan  = F/N At the point of impending motion: R= N 2 +F 2 max = N 2 +(  s N) 2 = N 1+  s 2 tan  s = F/N=  s N/N=  s  s is the angle of static friction  s

9. Gravity forces in inclined surfaces  s For equilibrium:

10. Example 9-4 Two blocks with masses m A = 20 kg and m B = 80 kg are connected with a flexible cable that passes over a frictionless pulley as shown in Fig. P9-4. The coefficient of friction between the blocks is 0.25. If motion of the blocks is impending, determine the coefficient of friction between block B and the inclined surface and the tension in the cable between the two blocks.

11. Solution  F y = A n - 196.14 cos(35  ) = 0 A n = 160.67 N For impending motion of the block A: (A f =  s. A n ) +   F x = -T + 0.25(160.67) + 196.14 sin 35  = 0 T = 152.67 N Free body diagram for block A

12. Free body diagram for block B: W B = m B. g = 80 (9.807) = 784.56 N Block B weight:  F y = B n - 160.67 - 784.56 cos 35  = 0 B n = 803.34 N  = B f /B n = 257.17/803.34 = 0.320 B f = 257.17 N  F x = - B f - 40.17 - 152.67 + 784.56 sin 35  = 0

13. Class Assignment: Exercise set P9-1 please submit to TA at the end of the lecture Determine the horizontal force P required to start moving the 250-lb block shown in Fig. P9-1 up the inclined surface. The coefficient of friction between the inclined surface and the block is  s = 0.30. P = 265.29 lb  265.3b

14. Solution For impending motion: F f =  F n = 0.30F n  F y = F n cos 30  - F f sin 30  -W = F n cos 30  - 0.30F n sin 30  - 250 = 0 Free body diagram for the block: F n = 349.15 lb  F x = P - F n sin 30  - F f cos 30  = P - 349.15 sin 30  - 0.30 (349.15) cos 30  = 0 P = 265.29 lb  265.3b

15. Class Assignment: Exercise set P9-36 please submit to TA at the end of the lecture The masses of blocks A and B of Fig. P9-36 are m A = 40 kg and m B = 85 kg. If the coefficient of friction is 0.25 for both surfaces, determine the force P required to cause impending motion of block B. P  935 N

16. Solution From a free-body diagram for the block A when motion is impending: A f = A f (max) =  A A n = 0.25 A n W A = m A g = 40 (9.807) = 392.28 N  F y = A n cos 45  + A f sin 45  - W A = A n cos 45  + 0.25 A n sin 45  - 392.28 = 0 A n = 443.81 N A f = 0.25 A n = 110.95 N

17. From a free-body diagram for the block B when motion is impending:  F x = P cos 20  -W B sin 45  -A f -B f = P cos 20  -833.60 sin 45  -110.95-0.25B n = 0 W B = m B g = 85 (9.807) = 833.60 N B f = B f (max) =  B n = 0.25B n  F y = P sin 20  - W B cos 45  - A n + B n = P sin 20  - 833.60 cos 45  - 443.81 + B n = 0 P  935 N

18. Class Assignment: Exercise set P9-3 please submit to TA at the end of the lecture Workers are pulling a 400 lb crate up an incline as shown in Fig. P9-3. The coefficient of friction between the crate and the surface is 0.20, and the rope on which the workers are pulling is horizontal. (a) Determine the force P that the workers must exert to start sliding the crate up the incline. (b) If one of the workers lets go of the rope for a moment, determine the minimum force the other workers must exert to keep the crate from sliding back down the incline. a)P=197.8 lb b)P=25.8lb

19. A 120 lb girl is walking up a 48-lb uniform beam as shown in Fig. P9-21. Determine how far up the beam the girl can walk before the beam starts to slip if (a)The coefficient of friction is 0.20 at all surfaces (b)The coefficient of friction at the bottom end of the beam is increased to 0.40 by placing a piece of rubber between the beam and the floor. a)x = 3.21 ft b)x = 6.26 ft Class Assignment: Exercise set P9-3 please submit to TA at the end of the lecture

20. Solution (a) For a free-body diagram for the beam when motion is impending: A r =  A n = 0.2A n B f =  B n = 0.2B n  = tan -1 6/8 = 36.87   F h = B n sin 36.87  - 0.2B n cos 36.87  - 0.2A n = 0  +   F y = B n cos 36.87  + 0.2B n sin 36.87  + A n - 120 - 48 = 0  Solving yields: A n = 118.46 lb; B n = 53.85 lb +  M A = 48(6) cos 36.87  + 120(x) cos 36.87  - 53.85(10) = 0 x = 3.209 ft  3.21 ft

21. (b)For a free-body diagram for the beam when motion is impending: A r =  A n = 0.4A n B f =  B n = 0.2B n +   F h = B n sin 36.87  - 0.2B n cos 36.87  - 0.4A n = 0 +   F y = B n cos 36.87  + 0.2B n sin 36.87  + A n - 120 - 48 = 0 Solving yields:A n = 91.49 lb; B n = 83.17 lb +  M A = 48(6) cos 36.87  + 120(x) cos 36.87  - 83.17(10) = 0 x = 6.264 ft  6.26 ft