he cylinder is subjected to a load that has a weight W .

The mine car and its contents have a total mass of 6 Mg and a center of gravity at G . If the coefficient of static friction between the wheels and the tracks is ms = 0.4 when the wheels are locked, find the normal force acting on the front wheels at B and the rear wheels at A when the brakes at both A and B are locked. Does the car move?

Determine the maximum force P the connection can support so that no slipping occurs between the plates. There are four bolts used for the connection and each is tightened so that it is subjected to a tension of 4 kN. The coefficient of static friction between the plates is ms = 0.4.

The winch on the truck is used to hoist the garbage bin onto the bed of the truck. If the loaded bin has a weight of 8500 lb and center of gravity at G , determine the force in the cable needed to begin the lift. The coefficients of static friction at A and B are mA = 0.3 and mB = 0.2, respectively. Neglect the height of the support at A .

The tractor has a weight of 4500 lb with center of gravity at G . The driving traction is developed at the rear wheels B , while the front wheels at A are free to roll. If the coefficient of static friction between the wheels at B and the ground is ms = 0.5, determine if it is possible to pull at P = 1200 lb without causing the wheels at B to slip or the front wheels at A to lift off the groun

The ladder has a uniform weight of 80 lb and rests against the smooth wall at B . If the coefficient of static friction at A is mA = 0.4 , determine if the ladder will slip. Take u = 60 .

The ladder has a uniform weight of 80 lb and rests against the wall at B. If the coefficient of static friction at A and B is \(\mu = 0.4\), determine the smallest angle \(\theta\) at which the ladder will not slip.

The block brake consists of a pin-connected lever and friction block at B. The coefficient of static friction between the wheel and the lever is \(\mu_s = 0.3\), and a torque of \(5\ N \cdot m\) is applied to the wheel. Determine if the brake can hold the wheel stationary when the force applied to the lever is (a) P = 30 N, (b) P = 70 N.

The block brake consists of a pin-connected lever and friction block at B. The coefficient of static friction between the wheel and the lever is \(\mu_s = 0.3\), and a torque of \(5\ N \cdot m\) is applied to the wheel. Determine if the brake can hold the wheel stationary when the force applied to the lever is (a) P = 30 N, (b) P = 70 N.

The block brake is used to stop the wheel from rotating when the wheel is subjected to a couple moment \(\mathbf{M}_0\). If the coefficient of static friction between the wheel and the block is \(\mu_s\), determine the smallest force P that should be applied.

Show that the brake in Prob. 8–9 is self-locking, i.e., \(P\ \leq\ 0\), provided \(b/c\ \leq\ \mu_s\).

The block brake is used to stop the wheel from rotating when the wheel is subjected to a couple moment \(\mathbf{M}_0\). If the coefficient of static friction between the wheel and the block is \(\mu_s\), determine the smallest force P that should be applied.

If a torque of \(M = 300\ N \cdot m\) is applied to the flywheel, determine the force that must be developed in the hydraulic cylinder CD to prevent the flywheel from rotating. The coefficient of static friction between the friction pad at B and the flywheel is \(\mu_s = 0.4\).

The cam is subjected to a couple moment of \(5\ N \cdot m\) Determine the minimum force P that should be applied to the follower in order to hold the cam in the position shown. The coefficient of static friction between the cam and the follower is \(\mu = 0.4\). The guide at A is smooth.

Determine the maximum weight W the man can lift with constant velocity using the pulley system, without and then with the “leading block” or pulley at A. The man has a weight of 200 lb and the coefficient of static friction between his feet and the ground is \(\mu_s = 0.6\).

The car has a mass of 1.6 Mg and center of mass at G. If the coefficient of static friction between the shoulder of the road and the tires is \(\mu_s = 0.4\), determine the greatest slope \(\theta\) the shoulder can have without causing the car to slip or tip over if the car travels along the shoulder at constant velocity.

The uniform dresser has a weight of 90 lb and rests on a tile floor for which \(\mu_s = 0.25\). If the man pushes on it in the horizontal direction \(\theta = 0^{\circ}\), determine the smallest magnitude of force F needed to move the dresser. Also, if the man has a weight of 150 lb, determine the smallest coefficient of static friction between his shoes and the floor so that he does not slip.

The uniform dresser has a weight of 90 lb and rests on a tile floor for which \(\mu_s = 0.25\). If the man pushes on it in the direction \(\theta = 30^{\circ}\), determine the smallest magnitude of force F needed to move the dresser. Also, if the man has a weight of 150 lb, determine the smallest coefficient of static friction between his shoes and the floor so that he does not slip.

The 5-kg cylinder is suspended from two equal length cords. The end of each cord is attached to a ring of negligible mass that passes along a horizontal shaft. If the rings can be separated by the greatest distance d = 400 mm and still support the cylinder, determine the coefficient of static friction between each ring and the shaft.

The 5-kg cylinder is suspended from two equal length cords. The end of each cord is attached to a ring of negligible mass that passes along a horizontal shaft. If the coefficient of static friction between each ring and the shaft is \(\mu_s = 0.5\), determine the greatest distance d by which the rings can be separated and still support the cylinder.

The board can be adjusted vertically by tilting it up and sliding the smooth pin A along the vertical guide G. When placed horizontally, the bottom C then bears along the edge of the guide, where \(\mu_s = 0.4\). Determine the largest dimension d which will support any applied force F without causing the board to slip downward.

The uniform pole has a weight W and length L. Its end B is tied to a supporting cord, and end A is placed against the wall, for which the coefficient of static friction is \(\mu_s\). Determine the largest angle \(\theta\) at which the pole can be placed without slipping.

If the clamping force is F = 200 N and each board has a mass of 2 kg, determine the maximum number of boards the clamp can support. The coefficient of static friction between the boards is \(\mu_s = 0.3\), and the coefficient of static friction between the boards and the clamp is \(\mu_s’ = 0.45\).

A 35-kg disk rests on an inclined surface for which \(\mu_s = 0.2\). Determine the maximum vertical force P that may be applied to bar AB without causing the disk to slip at C. Neglect the mass of the bar.

The man has a weight of 200 lb, and the coefficient of static friction between his shoes and the floor is \(\mu_s = 0.5\). Determine where he should position his center of gravity G at d in order to exert the maximum horizontal force on the door. What is this force?

The crate has a weight of W = 150 lb, and the coefficients of static and kinetic friction are \(\mu_s = 0.3\) and \(\mu_k = 0.2\), respectively. Determine the friction force on the floor if \(\theta = 30^{\circ}\) and P = 200 lb.

The crate has a weight of W = 350 lb, and the coefficients of static and kinetic friction are \(\mu_s = 0.3\) and \(\mu_k = 0.2\), respectively. Determine the friction force on the floor if \(\theta = 45^{\circ}\) and P = 100 lb.

The crate has a weight W and the coefficient of static friction at the surface is \(\mu_s = 0.3\). Determine the orientation of the cord and the smallest possible force P that has to be applied to the cord so that the crate is on the verge of moving.

If the coefficient of static friction between the man’s shoes and the pole is \(\mu_s = 0.6\), determine the minimum coefficient of static friction required between the belt and the pole at A in order to support the man. The man has a weight of 180 lb and a center of gravity at G.

The friction pawl is pinned at A and rests against the wheel at B. It allows freedom of movement when the wheel is rotating counterclockwise about C. Clockwise rotation is prevented due to friction of the pawl which tends to bind the wheel. If \((\mu_s)_B = 0.6\), determine the design angle \(\theta\) which will prevent clockwise motion for any value of applied moment M. Hint: Neglect the weight of the pawl so that it becomes a two-force member.

If \(\theta = 30^{\circ}\), determine the minimum coefficient of static friction at A and B so that equilibrium of the supporting frame is maintained regardless of the mass of the cylinder C. Neglect the mass of the rods.

If the coefficient of static friction at A and B is \(\mu_s = 0.6\), determine the maximum angle \(\theta\) so that the frame remains in equilibrium, regardless of the mass of the cylinder. Neglect the mass of the rods.

The semicylinder of mass m and radius r lies on the rough inclined plane for which \(\phi = 10^{\circ}\) and the coefficient of static friction is \(\mu_s = 0.3\). Determine if the semicylinder slides down the plane, and if not, find the angle of tip \(\theta\) of its base AB.

The semicylinder of mass m and radius r lies on the rough inclined plane. If the inclination \(\phi = 15^{\circ}\), determine the smallest coefficient of static friction which will prevent the semicylinder from slipping.

The coefficient of static friction between the 150-kg crate and the ground is \(\mu_s = 0.3\), while the coefficient of static friction between the 80-kg man’s shoes and the ground is \(\mu_s’ = 0.4\). Determine if the man can move the crate.

If the coefficient of static friction between the crate and the ground in Prob. 8–34 is \(\mu_s = 0.3\), determine the minimum coefficient of static friction between the man’s shoes and the ground so that the man can move the crate.

The rod has a weight W and rests against the floor and wall for which the coefficients of static friction are \(\mu_A\) and \(\mu_B\), respectively. Determine the smallest value of \(\theta\) for which the rod will not move.

The 80-lb boy stands on the beam and pulls on the cord with a force large enough to just cause him to slip. If \((\mu_s)_D = 0.4\) between his shoes and the beam, determine the reactions at A and B. The beam is uniform and has a weight of 100 lb. Neglect the size of the pulleys and the thickness of the beam.

The 80-lb boy stands on the beam and pulls with a force of 40 lb. If \((\mu_s)_D = 0.4\), determine the frictional force between his shoes and the beam and the reactions at A and B. The beam is uniform and has a weight of 100 lb. Neglect the size of the pulleys and the thickness of the beam.

Determine the smallest force the man must exert on the rope in order to move the 80-kg crate. Also, what is the angle \(\theta\) at this moment? The coefficient of static friction between the crate and the floor is \(\mu_s = 0.3\).

Two blocks A and B have a weight of 10 lb and 6 lb, respectively. They are resting on the incline for which the coefficients of static friction are \(\mu_A = 0.15\) and \(\mu_B = 0.25\). Determine the incline angle \(\theta\) for which both blocks begin to slide. Also find the required stretch or compression in the connecting spring for this to occur. The spring has a stiffness of k = 2 lb/ft.

Two blocks A and B have a weight of 10 lb and 6 lb, respectively. They are resting on the incline for which the coefficients of static friction are \(\mu_A = 0.15\) and \(\mu_B = 0.25\). Determine the angle \(\theta\) which will cause motion of one of the blocks. What is the friction force under each of the blocks when this occurs? The spring has a stiffness of k = 2 lb/ft and is originally unstretched.

The friction hook is made from a fixed frame and a cylinder of negligible weight. A piece of paper is placed between the wall and the cylinder. If \(\theta = 20^{\circ}\), determine the smallest coefficient of static friction \(\mu\) at all points of contact so that any weight W of paper \(\rho\) can be held.

The uniform rod has a mass of 10 kg and rests on the inside of the smooth ring at B and on the ground at A. If the rod is on the verge of slipping, determine the coefficient of static friction between the rod and the ground.

The rings A and C each weigh W and rest on the rod, which has a coefficient of static friction of \(\mu_s\). If the suspended ring at B has a weight of 2W, determine the largest distance d between A and C so that no motion occurs. Neglect the weight of the wire. The wire is smooth and has a total length of l.

The three bars have a weight of \(W_A = 20\ lb\), \(W_B = 40\ lb\), and \(W_C = 60\ lb\), respectively. If the coefficients of static friction at the surfaces of contact are as shown, determine the smallest horizontal force P needed to move block A.

The beam AB has a negligible mass and thickness and is subjected to a triangular distributed loading. It is supported at one end by a pin and at the other end by a post having a mass of 50 kg and negligible thickness. Determine the minimum force P needed to move the post. The coefficients of static friction at B and C are \(\mu_B = 0.4\) and \(\mu_C = 0.2\), respectively.

The beam AB has a negligible mass and thickness and is subjected to a triangular distributed loading. It is supported at one end by a pin and at the other end by a post having a mass of 50 kg and negligible thickness. Determine the two coefficients of static friction at B and at C so that when the magnitude of the applied force is increased to P = 150 N, the post slips at both B and C simultaneously.

The beam AB has a negligible mass and thickness and is subjected to a force of 200 N. It is supported at one end by a pin and at the other end by a spool having a mass of 40 kg. If a cable is wrapped around the inner core of the spool, determine the minimum cable force P needed to move the spool. The coefficients of static friction at B and D are \(\mu_B = 0.4\) and \(\mu_D = 0.2\), respectively.

If each box weighs 150 lb, determine the least horizontal force P that the man must exert on the top box in order to cause motion. The coefficient of static friction between the boxes is \(\mu_s = 0.5\), and the coefficient of static friction between the box and the floor is \(\mu_s’ = 0.2\).

If each box weighs 150 lb, determine the least horizontal force P that the man must exert on the top box in order to cause motion. The coefficient of static friction between the boxes is \(\mu_s = 0.65\), and the coefficient of static friction between the box and the floor is \(\mu_s’ = 0.35\).

The block of weight W is being pulled up the inclined plane of slope \(\alpha\) using a force P. If P acts at the angle \(\phi\) as shown, show that for slipping to occur, \(P = W\ \sin(\alpha + \theta)\ /\ \cos(\phi - \theta)\), where \(\theta\) is the angle of friction; \(\theta = \tan^{-1}\ \mu\).

Determine the angle \(\phi\) at which P should act on the block so that the magnitude of P is as small as possible to begin pulling the block up the incline. What is the corresponding value of P ? The block weighs W and the slope \(\alpha\) is known.

The wheel weighs 20 lb and rests on a surface for which \(\mu_B = 0.2.\ A\) cord wrapped around it is attached to the top of the 30-lb homogeneous block. If the coefficient of static friction at D is \(\mu_D = 0.3\), determine the smallest vertical force that can be applied tangentially to the wheel which will cause motion to impend.

The uniform beam has a weight W and length 4a. It rests on the fixed rails at A and B. If the coefficient of static friction at the rails is \(\mu_s\), determine the horizontal force P, applied perpendicular to the face of the beam, which will cause the beam to move.

Determine the greatest angle \(\theta\) so that the ladder does not slip when it supports the 75-kg man in the position shown. The surface is rather slippery, where the coefficient of static friction at A and B is \(\mu_s = 0.3\).

The uniform 6-kg slender rod rests on the top center of the 3-kg block. If the coefficients of static friction at the points of contact are \(\mu_A = 0.4\), \(\mu_B = 0.6\), and \(\mu_C = 0.3\), determine the largest couple moment M which can be applied to the rod without causing motion of the rod.

The disk has a weight W and lies on a plane that has a coefficient of static friction \(\mu\). Determine the maximum height h to which the plane can be lifted without causing the disk to slip.

Determine the largest angle \(\theta\) that will cause the wedge to be self-locking regardless of the magnitude of horizontal force P applied to the blocks. The coefficient of static friction between the wedge and the blocks is \(\mu_s = 0.3\). Neglect the weight of the wedge.

If the beam AD is loaded as shown, determine the horizontal force P which must be applied to the wedge in order to remove it from under the beam. The coefficients of static friction at the wedge’s top and bottom surfaces are \(\mu_{CA} = 0.25\) and \(\mu_{CB} = 0.35\), respectively. If P = 0, is the wedge self-locking? Neglect the weight and size of the wedge and the thickness of the beam.

The wedge has a negligible weight and a coefficient of static friction \(\mu_s = 0.35\) with all contacting surfaces. Determine the largest angle \(\theta\) so that it is “self-locking.” This requires no slipping for any magnitude of the force P applied to the joint.

If the spring is compressed 60 mm and the coefficient of static friction between the tapered stub S and the slider A is \(\mu_{SA} = 0.5\), determine the horizontal force P needed to move the slider forward. The stub is free to move without friction within the fixed collar C. The coefficient of static friction between A and surface B is \(\mu_{AB} = 0.4\). Neglect the weights of the slider and stub.

If P = 250 N, determine the required minimum compression in the spring so that the wedge will not move to the right. Neglect the weight of A and B. The coefficient of static friction for all contacting surfaces is \(\mu_s = 0.35\). Neglect friction at the rollers.

Determine the minimum applied force P required to move wedge A to the right. The spring is compressed a distance of 175 mm. Neglect the weight of A and B. The coefficient of static friction for all contacting surfaces is \(\mu_s = 0.35\). Neglect friction at the rollers.

Determine the largest weight of the wedge that can be placed between the 8-lb cylinder and the wall without upsetting equilibrium. The coefficient of static friction at A and C is \(\mu_s = 0.5\) and at B, \(\mu_s’ = 0.6\).

The coefficient of static friction between wedges B and C is \(\mu_s = 0.6\) and between the surfaces of contact B and A and C and D, \(\mu_s’ = 0.4\). If the spring is compressed 200 mm when in the position shown, determine the smallest force P needed to move wedge C to the left. Neglect the weight of the wedges.

The coefficient of static friction between the wedges B and C is \(\mu_s = 0.6\) and between the surfaces of contact B and A and C and D, \(\mu_s’ = 0.4\). If P = 50 N, determine the smallest allowable compression of the spring without causing wedge C to move to the left. Neglect the weight of the wedges.

If couple forces of F = 10 lb are applied perpendicular to the lever of the clamp at A and B, determine the clamping force on the boards. The single square-threaded screw of the clamp has a mean diameter of 1 in. and a lead of 0.25 in. The coefficient of static friction is \(\mu_s = 0.3\).

If the clamping force on the boards is 600 lb, determine the required magnitude of the couple forces that must be applied perpendicular to the lever AB of the clamp at A and B in order to loosen the screw. The single square threaded screw has a mean diameter of 1 in. and a lead of 0.25 in. The coefficient of static friction is \(\mu_s = 0.3\).

The column is used to support the upper floor. If a force F = 80 N is applied perpendicular to the handle to tighten the screw, determine the compressive force in the column. The square-threaded screw on the jack has a coefficient of static friction of \(\mu_s = 0.4\), mean diameter of 25 mm, and a lead of 3 mm.

If the force F is removed from the handle of the jack in Prob. 8–69, determine if the screw is self-locking.

If the clamping force at G is 900 N, determine the horizontal force F that must be applied perpendicular to the handle of the lever at E. The mean diameter and lead of both single square-threaded screws at C and D are 25 mm and 5 mm, respectively. The coefficient of static friction is \(\mu_s = 0.3\).

If a horizontal force of F = 50 N is applied perpendicular to the handle of the lever at E, determine the clamping force developed at G. The mean diameter and lead of the single square-threaded screw at C and D are 25 mm and 5 mm, respectively. The coefficient of static friction is \(\mu_s = 0.3\).

A turnbuckle, similar to that shown in Fig. 8–17, is used to tension member AB of the truss. The coefficient of the static friction between the square threaded screws and the turnbuckle is \(\mu_s = 0.5\). The screws have a mean radius of 6 mm and a lead of 3 mm. If a torque of \(\mathbf{M} = 10\ N \cdot m\) is applied to the turnbuckle, to draw the screws closer together, determine the force in each member of the truss. No external forces act on the truss.

A turnbuckle, similar to that shown in Fig. 8–17, is used to tension member AB of the truss. The coefficient of the static friction between the square-threaded screws and the turnbuckle is \(\mu_s = 0.5\). The screws have a mean radius of 6 mm and a lead of 3 mm. Determine the torque M which must be applied to the turnbuckle to draw the screws closer together, so that the compressive force of 500 N is developed in member BC.

The shaft has a square-threaded screw with a lead of 8 mm and a mean radius of 15 mm. If it is in contact with a plate gear having a mean radius of 30 mm, determine the resisting torque M on the plate gear which can be overcome if a torque of \(7\ N \cdot m\) is applied to the shaft. The coefficient of static friction at the screw is \(\mu_B = 0.2\). Neglect friction of the bearings located at A and B.

The square-threaded screw has a mean diameter of 20 mm and a lead of 4 mm. If the weight of the plate A is 5 lb, determine the smallest coefficient of static friction between the screw and the plate so that the plate does not travel down the screw when the plate is suspended as shown.

The fixture clamp consist of a square-threaded screw having a coefficient of static friction of \(\mu_s = 0.3\), mean diameter of 3 mm, and a lead of 1 mm. The five points indicated are pin connections. Determine the clamping force at the smooth blocks D and E when a torque of \(M = 0.08\ N \cdot m\) is applied to the handle of the screw.

The braking mechanism consists of two pinned arms and a square-threaded screw with left- and right-hand threads. Thus when turned, the screw draws the two arms together. If the lead of the screw is 4 mm, the mean diameter 12 mm, and the coefficient of static friction is \(\mu_s = 0.35\), determine the tension in the screw when a torque of \(5\ N \cdot m\) is applied to tighten the screw. If the coefficient of static friction between the brake pads A and B and the circular shaft is \(\mu_s’ = 0.5\), determine the maximum torque M the brake can resist.

If a horizontal force of P = 100 N is applied perpendicular to the handle of the lever at A, determine the compressive force F exerted on the material. Each single square-threaded screw has a mean diameter of 25 mm and a lead of 7.5 mm. The coefficient of static friction at all contacting surfaces of the wedges is \(\mu_s = 0.2\), and the coefficient of static friction at the screw is \(\mu_s’ = 0.15\).

Determine the horizontal force P that must be applied perpendicular to the handle of the lever at A in order to develop a compressive force of 12 kN on the material. Each single square-threaded screw has a mean diameter of 25 mm and a lead of 7.5 mm. The coefficient of static friction at all contacting surfaces of the wedges is \(\mu_s = 0.2\), and the coefficient of static friction at the screw is \(\mu_s’ = 0.15\).

Determine the clamping force on the board A if the screw of the “C” clamp is tightened with a twist of \(M = 8\ N \cdot m\). The single square-threaded screw has a mean radius of 10 mm, a lead of 3 mm, and the coefficient of static friction is \(\mu_s = 0.35\).

If the required clamping force at the board A is to be 50 N, determine the torque M that must be applied to the handle of the “C” clamp to tighten it down. The single square-threaded screw has a mean radius of 10 mm, a lead of 3 mm, and the coefficient of static friction is \(\mu_s = 0.35\).

A cylinder having a mass of 250 kg is to be supported by the cord that wraps over the pipe. Determine the smallest vertical force F needed to support the load if the cord passes (a) once over the pipe, \(\beta = 180^{\circ}\), and (b) two times over the pipe, \(\beta = 540^{\circ}\). Take \(\mu_s = 0.2\).

A cylinder having a mass of 250 kg is to be supported by the cord that wraps over the pipe. Determine the largest vertical force F that can be applied to the cord without moving the cylinder. The cord passes (a) once over the pipe, \(\beta = 180^{\circ}\), and (b) two times over the pipe, \(\beta = 540^{\circ}\). Take \(\mu_s = 0.2\).

A “hawser” is wrapped around a fixed “capstan” to secure a ship for docking. If the tension in the rope, caused by the ship, is 1500 lb, determine the least number of complete turns the rope must be rapped around the capstan in order to prevent slipping of the rope. The greatest horizontal force that a longshoreman can exert on the rope is 50 lb. The coefficient of static friction is \(\mu_s = 0.3\).

A force of P = 25 N is just sufficient to prevent the 20-kg cylinder from descending. Determine the required force P to begin lifting the cylinder. The rope passes over a rough peg with two and half turns.

The 20-kg cylinder A and 50-kg cylinder B are connected together using a rope that passes around a rough peg two and a half turns. If the cylinders are on the verge of moving, determine the coefficient of static friction between the rope and the peg.

Determine the maximum and the minimum values of weight W which may be applied without causing the 50-lb block to slip. The coefficient of static friction between the block and the plane is \(\mu_s = 0.2\), and between the rope and the drum D \(\mu_s’ = 0.3\).

The truck, which has a mass of 3.4 Mg, is to be lowered down the slope by a rope that is wrapped around a tree. If the wheels are free to roll and the man at A can resist a pull of 300 N, determine the minimum number of turns the rope should be wrapped around the tree to lower the truck at a constant speed. The coefficient of kinetic friction between the tree and rope is \(\mu_k = 0.3\).

The smooth beam is being hoisted using a rope that is wrapped around the beam and passes through a ring at A as shown. If the end of the rope is subjected to a tension T and the coefficient of static friction between the rope and ring is \(\mu_s = 0.3\), determine the smallest angle of \(\theta\) for equilibrium.

The uniform concrete pipe has a weight of 800 lb and is unloaded slowly from the truck bed using the rope and skids shown. If the coefficient of kinetic friction between the rope and pipe is \(\mu_k = 0.3\), determine the force the worker must exert on the rope to lower the pipe at constant speed. There is a pulley at B, and the pipe does not slip on the skids. The lower portion of the rope is parallel to the skids.

The simple band brake is constructed so that the ends of the friction strap are connected to the pin at A and the lever arm at B. If the wheel is subjected to a torque of \(M = 80\ lb \cdot ft\), and the minimum force P = 20 lb is needed to apply to the lever to hold the wheel stationary, determine the coefficient of static friction between the wheel and the band.

The simple band brake is constructed so that the ends of the friction strap are connected to the pin at A and the lever arm at B. If the wheel is subjected to a torque of \(M = 80\ lb \cdot ft\), determine the smallest force P applied to the lever that is required to hold the wheel stationary. The coefficient of static friction between the strap and wheel is \(\mu_s = 0.5\).

A minimum force of P = 50 lb is required to hold the cylinder from slipping against the belt and the wall. Determine the weight of the cylinder if the coefficient of friction between the belt and cylinder is \(\mu_s = 0.3\) and slipping does not occur at the wall.

The cylinder weighs 10 lb and is held in equilibrium by the belt and wall. If slipping does not occur at the wall, determine the minimum vertical force P which must be applied to the belt for equilibrium. The coefficient of static friction between the belt and the cylinder is \(\mu_s = 0.25\).

A cord having a weight of 0.5 lb/ft and a total length of 10 ft is suspended over a peg P as shown. If the coefficient of static friction between the peg and cord is \(\mu_s = 0.5\), determine the longest length h which one side of the suspended cord can have without causing motion. Neglect the size of the peg and the length of cord draped over it.

Determine the smallest force P required to lift the 40-kg crate. The coefficient of static friction between the cable and each peg is \(\mu_s = 0.1\).

Show that the frictional relationship between the belt tensions, the coefficient of friction \(\mu\), and the angular contacts \(\alpha\) and \(\beta\) for the V-belt is \(T_2 = T_1 e^{\mu \beta / \sin (\alpha / 2)}\).

If a force of P = 200 N is applied to the handle of the bell crank, determine the maximum torque M that can be resisted so that the flywheel does not rotate clockwise. The coefficient of static friction between the brake band and the rim of the wheel is \(\mu_s = 0.3\).

A 10-kg cylinder D, which is attached to a small pulley B, is placed on the cord as shown. Determine the largest angle \(\theta\) so that the cord does not slip over the peg at C. The cylinder at E also has a mass of 10 kg, and the coefficient of static friction between the cord and the peg is \(\mu_s = 0.1\).

A V-belt is used to connect the hub A of the motor to wheel B. If the belt can withstand a maximum tension of 1200 N, determine the largest mass of cylinder C that can be lifted and the corresponding torqueM that must be supplied to A. The coefficient of static friction between the hub and the belt is \(\mu_s = 0.3\), and between the wheel and the belt is \(\mu_s’ = 0.20\). Hint: See Prob. 8–98.

The 20-kg motor has a center of gravity at G and is pin-connected at C to maintain a tension in the drive belt. Determine the smallest counterclockwise twist or torque M that must be supplied by the motor to turn the disk B if wheel A locks and causes the belt to slip over the disk. No slipping occurs at A. The coefficient of static friction between the belt and the disk is \(\mu_s = 0.3\).

Blocks A and B have a mass of 100 kg and 150 kg, respectively. If the coefficient of static friction between A and B and between B and C is \(\mu_s = 0.25\) and between the ropes and the pegs D and E \(\mu_s’ = 0.5\) determine the smallest force F needed to cause motion of block B if P = 30 N.

Determine the minimum coefficient of static friction \(\mu_s\) between the cable and the peg and the placement d of the 3-kN force for the uniform 100-kg beam to maintain equilibrium.

A conveyer belt is used to transfer granular material and the frictional resistance on the top of the belt is F = 500 N. Determine the smallest stretch of the spring attached to the moveable axle of the idle pulley B so that the belt does not slip at the drive pulley A when the torque M is applied. What minimum torque M is required to keep the belt moving? The coefficient of static friction between the belt and the wheel at A is \(\mu_s = 0.2\).

The belt on the portable dryer wraps around the drum D, idler pulley A, and motor pulley B. If the motor can develop a maximum torque of \(M = 0.80\ N \cdot m\), determine the smallest spring tension required to hold the belt from slipping. The coefficient of static friction between the belt and the drum and motor pulley is \(\mu_s = 0.3\).

The annular ring bearing is subjected to a thrust of 800 lb. Determine the smallest required coefficient of static friction if a torque of \(M = 15\ lb \cdot ft\) must be resisted to prevent the shaft from rotating.

The annular ring bearing is subjected to a thrust of 800 lb. If \(\mu_s = 0.35\), determine the torque M that must be applied to overcome friction.

The floor-polishing machine rotates at a constant angular velocity. If it has a weight of 80 lb. determine the couple forces F the operator must apply to the handles to hold the machine stationary. The coefficient of kinetic friction between the floor and brush is \(\mu_k = 0.3\). Assume the brush exerts a uniform pressure on the floor.

The shaft is supported by a thrust bearing A and a journal bearing B. Determine the torque M required to rotate the shaft at constant angular velocity. The coefficient of kinetic friction at the thrust bearing is \(\mu_k = 0.2\). Neglect friction at B.

The thrust bearing supports an axial load of P = 6 kN. If a torque of \(M = 150\ N \cdot m\) is required to rotate the shaft, determine the coefficient of static friction at the constant surface.

Assuming that the variation of pressure at the bottom of the pivot bearing is defined as \(p = p_0 (R_2/r)\), determine the torque M needed to overcome friction if the shaft is subjected to an axial force P. The coefficient of static friction is \(\mu_s\). For the solution, it is necessary to determine \(\rho_0\) in terms of P and the bearing dimensions \(R_1\) and \(R_2\).

The plate clutch consists of a flat plate A that slides over the rotating shaft S. The shaft is fixed to the driving plate gear B. If the gear C, which is in mesh with B, is subjected to a torque of \(M = 0.8\ N \cdot m\), determine the smallest force P, that must be applied via the control arm, to stop the rotation. The coefficient of static friction between the plates A and D is \(\mu_s = 0.4\). Assume the bearing pressure between A and D to be uniform.

The conical bearing is subjected to a constant pressure distribution at its surface of contact. If the coefficient of static friction is \(\mu_s\), determine the torque M required to overcome friction if the shaft supports an axial force P.

The pivot bearing is subjected to a pressure distribution at its surface of contact which varies as shown. If the coefficient of static friction is \(\mu\), determine the torque M required to overcome friction if the shaft supports an axial force P.

A 200-mm-diameter post is driven 3 m into sand for which \(\mu_s = 0.3\). If the normal pressure acting completely around the post varies linearly with depth as shown, determine the frictional torque M that must be overcome to rotate the post.

A beam having a uniform weight W rests on the rough horizontal surface having a coefficient of static friction \(\mu_s\). If the horizontal force P is applied perpendicular to the beam’s length, determine the location d of the point O about which the beam begins to rotate.

The collar fits loosely around a fixed shaft that has a radius of 2 in. If the coefficient of kinetic friction between the shaft and the collar is \(\mu_k = 0.3\), determine the force P on the horizontal segment of the belt so that the collar rotates counterclockwise with a constant angular velocity. Assume that the belt does not slip on the collar; rather, the collar slips on the shaft. Neglect the weight and thickness of the belt and collar. The radius, measured from the center of the collar to the mean thickness of the belt, is 2.25 in.

The collar fits loosely around a fixed shaft that has a radius of 2 in. If the coefficient of kinetic friction between the shaft and the collar is \(\mu_k = 0.3\), determine the force P on the horizontal segment of the belt so that the collar rotates clockwise with a constant angular velocity. Assume that the belt does not slip on the collar; rather, the collar slips on the shaft. Neglect the weight and thickness of the belt and collar. The radius, measured from the center of the collar to the mean thickness of the belt, is 2.25 in.

The pulley has a radius of 3 in. and fits loosely on the 0.5-in.-diameter shaft. If the loadings acting on the belt cause the pulley to rotate with constant angular velocity, determine the frictional force between the shaft and the pulley and compute the coefficient of kinetic friction. The pulley weighs 18 lb.

The pulley has a radius of 3 in. and fits loosely on the 0.5-in.-diameter shaft. If the loadings acting on the belt cause the pulley to rotate with constant angular velocity, determine the frictional force between the shaft and the pulley and compute the coefficient of kinetic friction. Neglect the weight of the pulley.

Determine the tension T in the belt needed to overcome the tension of 200 lb created on the other side. Also, what are the normal and frictional components of force developed on the collar bushing? The coefficient of static friction is \(\mu_s = 0.21\).

If a tension force T = 215 lb is required to pull the 200-lb force around the collar bushing, determine the coefficient of static friction at the contacting surface. The belt does not slip on the collar.

A pulley having a diameter of 80 mm and mass of 1.25 kg is supported loosely on a shaft having a diameter of 20 mm. Determine the torque M that must be applied to the pulley to cause it to rotate with constant motion. The coefficient of kinetic friction between the shaft and pulley is \(\mu_k = 0.4\). Also calculate the angle \(\theta\) which the normal force at the point of contact makes with the horizontal. The shaft itself cannot rotate.

The 5-kg skateboard rolls down the \(5^{\circ}\) slope at constant speed. If the coefficient of kinetic friction between the 12.5 mm diameter axles and the wheels is \(\mu_k = 0.3\), determine the radius of the wheels. Neglect rolling resistance of the wheels on the surface. The center of mass for the skateboard is at G.

The cart together with the load weighs 150 lb and has a center of gravity at G. If the wheels fit loosely on the 1.5-in. diameter axles, determine the horizontal force P required to pull the cart with constant velocity. The coefficient of kinetic friction between the axles and the wheels is \(\mu_k = 0.2\). Neglect rolling resistance of the wheels on the ground.

The trailer has a total weight of 850 lb and center of gravity at G which is directly over its axle. If the axle has a diameter of 1 in., the radius of the wheel is r = 1.5 ft, and the coefficient of kinetic friction at the bearing is \(\mu_k = 0.08\), determine the horizontal force P needed to pull the trailer.

The vehicle has a weight of 2600 lb and center of gravity at G. Determine the horizontal force P that must be applied to overcome the rolling resistance of the wheels. The coefficient of rolling resistance is 0.5 in. The tires have a diameter of 2.75 ft.

The tractor has a weight of 16 000 lb and the coefficient of rolling resistance is a = 2 in. Determine the force P needed to overcome rolling resistance at all four wheels and push it forward.

The hand cart has wheels with a diameter of 80 mm. If a crate having a mass of 500 kg is placed on the cart so that each wheel carries an equal load, determine the horizontal force P that must be applied to the handle to overcome the rolling resistance. The coefficient of rolling resistance is 2 mm. Neglect the mass of the cart.

The cylinder is subjected to a load that has a weight W. If the coefficients of rolling resistance for the cylinder’s top and bottom surfaces are \(a_A\) and \(a_B\), respectively, show that a horizontal force having a magnitude of \(P = [W(a_A + a_B )] / 2r\) is required to move the load and thereby roll the cylinder forward. Neglect the weight of the cylinder.

A large crate having a mass of 200 kg is moved along the floor using a series of 150-mm-diameter rollers for which the coefficient of rolling resistance is 3 mm at the ground and 7 mm at the bottom surface of the crate. Determine the horizontal force P needed to push the crate forward at a constant speed. Hint: Use the result of Prob. 8–131.