Chapter 8 FRICTION W A horizontal force P applied to a block will not at first cause it to move. This is because the friction force F balances P. P F As the magnitude of P increases, the magnitude of F increases until it reaches a maximum value Fm. If P is increased further, the magnitude of F drops to Fk and the block begins to slide. N F Equilibrium Motion Fm Fk P

W Equilibrium Motion F Fm P Fk F N P The forces Fm and Fk are proportional to the normal component N of the reaction of the surface. We have Fm = ms N Fk = mk N where ms is the coefficient of static friction and mk is the coefficient of kinetic friction. These coefficients depend on the nature and the condition of the surfaces in contact.

W It is sometimes convenient to replace the normal force N and the friction force F by their resultant R. As the friction force increases and reaches its maximum value Fm =msN, the angle f that R forms with the normal to the surface increases and reaches a maximum value fs, called the angle of static friction. P R N f F F = R sin f N = R cos f

F = R sin f tan fs = ms N = R cos f tan fk = mk W If motion actually takes place, the magnitude of F drops to Fk; similarly the angle f drops to a lower value fk , called the angle of kinetic friction. The coefficient of friction and the angle of friction are related by P R N f F F = R sin f N = R cos f tan fs = ms tan fk = mk

W P Frequired N The magnitude F of the friction force is equal to Fm = msN only if the body is about to slide. If motion is not impending, F and N should be considered as independent unknowns to be determined from the equilibrium equations. The value of F required to maintain equilibrium should be checked to insure that it does not exceed Fm.

W P Fm = msN N If motion is known to be impending, F has reached its maximum value Fm = msN , and this expression may be substituted for F in the equilibrium equations.

B A P C D In the analysis of wedges, two or more free-body diagrams are generally used to show each friction force and its correct sense.

W The analysis of square- threaded screws (frequently used in jacks, presses and other mechanisms) is reduced to the analysis of a block sliding on an incline by unwrapping the thread of the screw and showing it as a straight line. Q L R q fs q 2pr In doing this, r denotes the mean radius of the thread, L is the lead of the screw (the distance through which the screw advances in one turn), W is the load, and Qr is the torque exerted on the screw.

P P’ Dq P2 P1 q b O T1 T2 For a flat belt passing over a cylinder, it is important to determine the direction in which the belt slips or is about to slip. If the drum is rotating, the motion or impending motion of the belt should be determined relative to the rotating drum.

P P’ If the belt shown is about to slip to the right relative to the drum, the friction force will be directed to the left and the tension will be larger in the right-hand portion of the belt than the left-hand portion. Dq P2 P1 q b O T1 T2 Denoting T2 as the larger tension, ms as the coefficient of static friction, and b as the angle (in radians) subtended by the belt, the two tensions are related by T2 T1 T2 T1 ln = msb = e msb