Answer: If u = 30, determine the minimum coefficient of

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 \(\mu_{\mathrm{s}}=0.4\).

The tractor exerts a towing force T = 400 lb. Determine the normal reactions at each of the two front and two rear tires and the tractive frictional force F on each rear tire needed to pull the load forward at constant velocity. The tractor has a weight of 7500 lb and a center of gravity located at \(G_{T}\). An additinal weight of 600 lb is added to its front having a center of gravity at \(G_{A}\). Take \(\mu_{\mathrm{s}}=0.4\). The front wheels are free to roll.

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 \(\mu_{\mathrm{s}}=0.4\) 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?

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 \(\mu_{\mathrm{A}}=0.3\) and \(\mu_{\mathrm{B}}=0.2\), respectively. Neglect the height of the support at A.

The automobile has a mass of 2 Mg and center of mass at G. Determine the towing force F required to move the car if the back brakes are locked, and the front wheels are free to roll. Take \(\mu_{\mathrm{s}}=0.3\).

The automobile has a mass of 2 Mg and center of mass at G. Determine the towing force F required to move the car. Both the front and rear brakes are locked. Take \(\mu_{\mathrm{s}}=0.3\).

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\(\mu_{\mathrm{s}}=0.3\), 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 \(\mu_{\mathrm{s}}=0.3\), 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 pipe of weight W is to be pulled up the inclined plane of slope a using a force P. If P acts at an angle f, show that for slipping \(P=W \sin (\alpha+\theta) / \cos (\phi-\theta)\), where \(\theta\) is the angle of static friction; \(\theta=\tan ^{-1} \mu_{s}\).

Determine the angle \(\phi\) at which the applied force P should act on the pipe so that the magnitude of P is as small as possible for pulling the pipe up the incline. What is the corresponding value of P? The pipe weighs W and the slope a is known. Express the answer in terms of the angle of kinetic friction, \(\theta=\tan ^{-1} \mu_{k}\).

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 \(\mu_{\mathrm{s}}=0.6\).

The block brake is used to stop the wheel from rotating when the wheel is subjected to a couple moment \(M_{0}\). If the coefficient of static friction between the wheel and the block is \(\mu_{\mathrm{s}}\), 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 \(\mu_{\mathrm{s}}=0.4\).

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 \(\mu_{\mathrm{s}}=0.4\), determine the greatest slope u the shoulder can have without causing the car to slip or tip over if the car travels along the shoulder at constant velocity.

The log has a coefficient of state friction of \(\mu_{\mathrm{s}}=0.4\) with the ground and a weight of 40 lb/ft. If a man can pull on the rope with a maximum force of 80 lb, determine the greatest length l of log he can drag.

The 180-lb man climbs up the ladder and stops at the position shown after he senses that the ladder is on the verge of slipping. Determine the inclination \(\theta\) of the ladder if the coefficient of static friction between the friction pad A and the ground is \(\mu_{\mathrm{s}}=0.4\). Assume the wall at B is smooth. The center of gravity for the man is at G. Neglect the weight of the ladder.

The 180-lb man climbs up the ladder and stops at the position shown after he senses that the ladder is on the verge of slipping. Determine the coefficient of static friction between the friction pad at A and ground if the inclination of the ladder is \(\theta=60^{\circ}\) and the wall at B is smooth. The center of gravity for the man is at G. Neglect the weight of the ladder

The spool of wire having a weight of 300 lb rests on the ground at B and against the wall at A. Determine the force P required to begin pulling the wire horizontally off the spool. The coefficient of static friction between the spool and its points of contact is \(\mu_{\mathrm{s}}=0.25\).

The spool of wire having a weight of 300 lb rests on the ground at B and against the wall at A. Determine the normal force acting on the spool at A if P = 300 lb. The coefficient of static friction between the spool and the ground at B is \(\mu_{\mathrm{s}}=0.35\). The wall at A is smooth.

The ring has a mass of 0.5 kg and is resting on the surface of the table. In an effort to move the ring a normal force P from the finger is exerted on it. If this force is directed towards the ring’s center O as shown, determine its magnitude when the ring is on the verge of slipping at A. The coefficient of static friction at A is \(\mu_{\mathrm{A}}=0.2\) and at B, \(\mu_{\mathrm{B}}=0.3\).

A man attempts to support a stack of books horizontally by applying a compressive force of F = 120 N to the ends of the stack with his hands. If each book has a mass of 0.95 kg, determine the greatest number of books that can be supported in the stack. The coefficient of static friction between his hands and a book is \(\left(\mu_{s}\right)_{h}=0.6\) and between any two books \(\left(\mu_{s}\right)_{b}=0.4\).

The tongs are used to lift the 150-kg crate, whose center of mass is at G. Determine the least coefficient of static friction at the pivot blocks so that the crate can be lifted.

The beam is supported by a pin at A and a roller at B which has negligible weight and a radius of 15 mm. If the coefficient of static friction is\(\mu_{B}=\mu_{C}=0.3\), determine the largest angle \(\theta\) of the incline so that the roller does not slip for any force P applied to the beam.

The uniform thin pole has a weight of 30 lb and a length of 26 ft. If it is placed against the smooth wall and on the rough floor in the position d = 10 ft, will it remain in this position when it is released? The coefficient of static friction is \(\mu_{\mathrm{s}}=0.3\)

The uniform pole has a weight of 30 lb and a length of 26 ft. Determine the maximum distance d it can be placed from the smooth wall and not slip. The coefficient of static friction between the floor and the pole is \(\mu_{\mathrm{s}}=0.3\).

The block brake is used to stop the wheel from rotating when the wheel is subjected to a couple moment \(M_{0}\) = 360 N \(\cdot\) m. If the coefficient of static friction between the wheel and the block is \(\mu_{\mathrm{s}}=0.6\), determine the smallest force P that should be applied.

Solve Prob. 8–26 if the couple moment \(M_{0}\) is applied counterclockwise.

A worker walks up the sloped roof that is defined by the curve \(y=\left(5 e^{0.01 x}\right)\) ft, where x is in feet. Determine how high h he can go without slipping. The coefficient of static friction is \(\mu_{\mathrm{s}}=0.6\).

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 \(\left(\mu_{s}\right)_{B}=0.6\), determine the design angle u 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.

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 \(\mu_{\mathrm{A}}=0.15\) and \(\mu_{\mathrm{B}}=0.25\). Determine the incline angle u 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 \(\mu_{\mathrm{A}}=0.15\) and \(\mu_{\mathrm{B}}=0.25\). Determine the angle u 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.

Determine the smallest force P that must be applied in order to cause the 150-lb uniform crate to move. The coefficent of static friction between the crate and the floor is \(\mu_{\mathrm{s}}=0.5\).

The man having a weight of 200 lb pushes horizontally on the crate. If the coefficient of static friction between the 450-lb crate and the floor is \(\mu_{\mathrm{s}}=0.3\) and between his shoes and the floor is \(\mu_{\mathrm{s}}=0.6\), determine if he can move the crate.

The uniform hoop of weight W is subjected to the horizontal force P. Determine the coefficient of static friction between the hoop and the surface of A and B if the hoop is on the verge of rotating.

Determine the maximum horizontal force P that can be applied to the 30-lb hoop without causing it to rotate. The coefficient of static friction between the hoop and the surfaces A and B is \(\mu_{\mathrm{s}}=0.2\). Take r = 300 mm.

Determine the minimum force P needed to push the tube E up the incline. The force acts parallel to the plane, and the coefficients of static friction at the contacting surfaces are \(\mu_{A}=0.2, \mu_{B}=0.3 \text {, and } \mu_{C}=0.4\). The 100-kg roller and 40-kg tube each have a radius of 150 mm.

The coefficients of static and kinetic friction between the drum and brake bar are \(\mu_{s}=0.4 \text { and } \mu_{k}=0.3 \text {, }\), respectively. If M = 50 N \(\cdot\) m and P = 85 N, determine the horizontal and vertical components of reaction at the pin O. Neglect the weight and thickness of the brake. The drum has a mass of 25 kg.

The coefficient of static friction between the drum and brake bar is \(\mu_{\mathrm{s}}=0.4\). If the moment M = 35 N \(\cdot\) m, determine the smallest force P that needs to be applied to the brake bar in order to prevent the drum from rotating. Also determine the corresponding horizontal and vertical components of reaction at pin O. Neglect the weight and thickness of the brake bar. The drum has a mass of 25 kg.

Determine the smallest coefficient of static friction at both A and B needed to hold the uniform 100-lb bar in equilibrium. Neglect the thickness of the bar. Take \(\mu_{A}=\mu_{B}=\mu\).

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. Neglect the mass of the rods.

If the coefficient of static friction at A and B is \(\mu_{\mathrm{s}}=0.4\), determine the maximum angle \(\theta\) so that the frame remains in equilbrium, regardless of the mass of the cylinder. Neglect the mass of the rods.

The 100-kg disk rests on a surface for which \(\mu_{\mathrm{B}}=0.2\). Determine the smallest vertical force P that can be applied tangentially to the disk which will cause motion to impend.

Investigate whether the equilibrium can be maintained. The uniform block has a mass of 500 kg, and the coefficient of static friction is \(\mu_{\mathrm{s}}=0.3\).

The homogenous semicylinder has a mass of 20 kg and mass center at G. If force P is applied at the edge, and r = 300 mm, determine the angle u at which the semicylinder is on the verge of slipping. The coefficient of static friction between the plane and the cylinder is \(\mu_{\mathrm{s}}=0.3\). Also, what is the corresponding force P for this case?

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 \(\mu_{\mathrm{B}}=0.4\) and \(\mu_{\mathrm{C}}=0.2\), 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.

Crates A and B weigh 200 lb and 150 lb, respectively. They are connected together with a cable and placed on the inclined plane. If the angle u is gradually increased, determine u when the crates begin to slide. The coefficients of static friction between the crates and the plane are \(\mu_{\mathrm{A}}=0.25\) and \(\mu_{\mathrm{B}}=0.35\).

Two blocks A and B, each having a mass of 5 kg, are connected by the linkage shown. If the coefficient of static friction at the contacting surfaces is \(\mu_{\mathrm{s}}=0.5\), determine the largest force P that can be applied to pin C of the linkage without causing the blocks to move. Neglect the weight of the links.

The uniform crate has a mass of 150 kg. If the coefficient of static friction between the crate and the floor is \(\mu_{\mathrm{s}}=0.2\), determine whether the 85-kg man can move the crate. The coefficient of static friction between his shoes and the floor is \(\mu_{\mathrm{s}}=0.4\). Assume the man only exerts a horizontal force on the crate.

The uniform crate has a mass of 150 kg. If the coefficient of static friction between the crate and the floor is \(\mu_{\mathrm{s}}=0.2\), determine the smallest mass of the man so he can move the crate. The coefficient of static friction between his shoes and the floor is \(\mu_{\mathrm{s}}=0.45\). Assume the man exerts only a horizontal force on the crate.

Beam AB has a negligible mass and thickness, and supports the 200-kg uniform block. It is pinned at A and rests on the top of a post, having a mass of 20 kg and negligible thickness. Determine the minimum force P needed to move the post. The coefficients of static friction at B and C are \(\mu_{\mathrm{B}}=0.4\) and \(\mu_{\mathrm{C}}=0.2\), respectively.

Beam AB has a negligible mass and thickness, and supports the 200-kg uniform block. It is pinned at A and rests on the top of a post, having a mass of 20 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 = 300 N, the post slips at both B and C simultaneously.

Determine the smallest couple moment that can be applied to the 150-lb wheel that will cause impending motion. The uniform concrete block has a weight of 300 lb. The coefficients of static friction are \(\mu_{\mathrm{A}}=0.2\), \(\mu_{\mathrm{B}}=0.3\), and between the concrete block and the floor, \(\mu=0.4\).

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 \(\mu_{\mathrm{s}}=0.3\).

The wheel weighs 20 lb and rests on a surface for which \(\mu_{\mathrm{B}}=0.2\). 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 \(\mu_{\mathrm{D}}=0.3\), determine the smallest vertical force that can be applied tangentially to the wheel which will cause motion to impend.

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.

The man has a weight of 200 lb, and the coefficient of static friction between his shoes and the floor is \(\mu_{\mathrm{s}}=0.5\). 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?

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 \(\mu_{\mathrm{s}}\) = 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 \(\mu_{\mathrm{CA}}\) = 0.25 and \(\mu_{\mathrm{CB}}\) = 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 is used to level the member. Determine the horizontal force P that must be applied to begin to push the wedge forward. The coefficient of static friction between the wedge and the two surfaces of contact is \(\mu_{\mathrm{s}}\) = 0.2. Neglect the weight of the wedge.

The two blocks used in a measuring device have negligible weight. If the spring is compressed 5 in. when in the position shown, determine the smallest axial force P which the adjustment screw must exert on B in order to start the movement of B downward. The end of the screw is smooth and the coefficient of static friction at all other points of contact is \(\mu_{\mathrm{s}}\) = 0.3

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 \(\mu_{\mathrm{s}}\) = 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 \(\mu_{\mathrm{s}}\) = 0.35. Neglect friction at the rollers.

If the coefficient of static friction between all the surfaces of contact is ms, determine the force P that must be applied to the wedge in order to lift the block having a weight W.

Determine the smallest force P needed to lift the 3000-lb load. The coefficient of static friction between A and C and between B and D is \(\mu_{\mathrm{s}}\) = 0.3, and between A and B \(\mu’_{\mathrm{s}}\) = 0.4. Neglect the weight of each wedge.

Determine the reversed horizontal force ?P needed to pull out wedge A. The coefficient of static friction between A and C and between B and D is \(\mu_{\mathrm{s}}\) = 0.2, and between A and B \(\mu’_{\mathrm{s}}\) = 0.1. Neglect the weight of each wedge.

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 \(\mu_{\mathrm{s}}\) = 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 \(\mu_{\mathrm{s}}\) = 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 \(\mu_{\mathrm{s}}\) = 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 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 \(\mu_{\mathrm{s}}\) = 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 \(\mu_{\mathrm{s}}\) = 0.3.

Prove that the lead l must be less than 2\(\pi\)r\(\mu_{\mathrm{s}}\) for the jack screw shown in Fig. 8–15 to be “self-locking.”

The square-threaded bolt is used to join two plates together. If the bolt has a mean diameter of d = 20 mm and a lead of l = 3 mm, determine the smallest torque M required to loosen the bolt if the tension in the bolt is T = 40 kN. The coefficient of static friction between the threads and the bolt is \(\mu_{\mathrm{s}}\) = 0.15.

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 \(\mu_{\mathrm{B}}\) = 0.2. Neglect friction of the bearings located at A and B.

If couple forces of F = 35 N are applied to the handle of the machinist’s vise, determine the compressive force developed in the block. Neglect friction at the bearing A. The guide at B is smooth. The single square-threaded screw has a mean radius of 6 mm and a lead of 8 mm, and the coefficient of static friction is \(\mu_{\mathrm{s}}\)s = 0.27.

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 device is used to pull the battery cable terminal C from the post of a battery. If the required pulling force is 85 lb, determine the torque M that must be applied to the handle on the screw to tighten it. The screw has square threads, a mean diameter of 0.2 in., a lead of 0.08 in., and the coefficient of static friction is \(\mu_{\mathrm{s}}\) = 0.5.

Determine the clamping force on the board A if the screw is tightened with a torque of M = 8 N \(\cdot\) m. The square-threaded screw has a mean radius of 10 mm and a lead of 3 mm, and the coefficient of static friction is \(\mu_{\mathrm{s}}\) = 0.35.

If the required clamping force at the board A is to be 2 kN, determine the torque M that must be applied to the screw to tighten it down. The square-threaded screw has a mean radius of 10 mm and a lead of 3 mm, and the coefficient of static friction is \(\mu_{\mathrm{s}}\) = 0.35.

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 \(\mu_{\mathrm{s}}\) = 0.2, and the coefficient of static friction at the screw is \(\mu’_{\mathrm{s}}\) = 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 \(\mu_{\mathrm{s}}\) = 0.2, and the coefficient of static friction at the screw is \(\mu’_{\mathrm{s}}\) = 0.15.

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 \(\mu_{\mathrm{s}}\) = 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 \(\mu_{\mathrm{s}}\) = 0.2.

A 180-lb farmer tries to restrain the cow from escaping by wrapping the rope two turns around the tree trunk as shown. If the cow exerts a force of 250 lb on the rope, determine if the farmer can successfully restrain the cow. The coefficient of static friction between the rope and the tree trunk is \(\mu_{\mathrm{s}}\) = 0.15, and between the farmer’s shoes and the ground \(\mu’_{\mathrm{s}}\) = 0.3.

The 100-lb boy at A is suspended from the cable that passes over the quarter circular cliff rock. Determine if it is possible for the 185-lb woman to hoist him up; and if this is possible, what smallest force must she exert on the horizontal cable? The coefficient of static friction between the cable and the rock is \(\mu_{\mathrm{s}}\) = 0.2, and between the shoes of the woman and the ground \(\mu’_{\mathrm{s}}\) = 0.8.

The 100-lb boy at A is suspended from the cable that passes over the quarter circular cliff rock. What horizontal force must the woman at A exert on the cable in order to let the boy descend at constant velocity? The coefficients of static and kinetic friction between the cable and the rock are \(\mu_{\mathrm{s}}\) = 0.4 and \(\mu_{\mathrm{l}}\) = 0.35, respectively.

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 \(\mu_{\mathrm{k}}\) = 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

A cable is attached to the 20-kg plate B, passes over a fixed peg at C, and is attached to the block at A. Using the coefficients of static friction shown, determine the smallest mass of block A so that it will prevent sliding motion of B down the plane.

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 \(\mu_{\mathrm{s}}\) = 0.3, determine the smallest angle of \(\theta\) for equilibrium

The boat has a weight of 500 lb and is held in position off the side of a ship by the spars at A and B. A man having a weight of 130 lb gets in the boat, wraps a rope around an overhead boom at C, and ties it to the end of the boat as shown. If the boat is disconnected from the spars, determine the minimum number of half turns the rope must make around the boom so that the boat can be safely lowered into the water at constant velocity. Also, what is the normal force between the boat and the man? The coefficient of kinetic friction between the rope and the boom is \(\mu_{\mathrm{s}}\) = 0.15. Hint: The problem requires that the normal force between the man’s feet and the boat be as small as possible.

Determine the force P that must be applied to the handle of the lever so that the wheel is on the verge of turning if M = 300 N \(\cdot\) m. The coefficient of static friction between the belt and the wheel is \(\mu_{\mathrm{s}}\) = 0.3.

If a force of P = 30 N is applied to the handle of the lever, determine the largest couple moment M that can be resisted so that the wheel does not turn. The coefficient of static friction between the belt and the wheel is \(\mu_{\mathrm{s}}\) = 0.3.

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 \(\mu_{\mathrm{s}}\) = 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 \(\mu_{\mathrm{s}}\) = 0.25.

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 \(\mu_{\mathrm{s}}\) = 0.2, and between the rope and the drum D is \(\mu’_{\mathrm{s}}\) = 0.3

Granular material, having a density of 1.5 Mg/\(m^{3}\) , is transported on a conveyor belt that slides over the fixed surface, having a coefficient of kinetic friction of \(\mu_{\mathrm{k}}\) = 0.3. Operation of the belt is provided by a motor that supplies a torque M to wheel A. The wheel at B is free to turn, and the coefficient of static friction between the wheel at A and the belt is \(\mu_{\mathrm{A}}\) = 0.4. If the belt is subjected to a pretension of 300 N when no load is on the belt, determine the greatest volume V of material that is permitted on the belt at any time without allowing the belt to stop. What is the torque M required to drive the belt when it is subjected to this maximum load?

Show that the frictional relationship between the belt tensions, the coefficient of friction m, and the angular contacts a and b for the V-belt is \(T_{2}=T_{1} e^{\mu \beta / \sin (\alpha / 2)}\).

The wheel is subjected to a torque of M = 50 N \(\cdot\) m. If the coefficient of kinetic friction between the band brake and the rim of the wheel is \(\mu_{\mathrm{k}}\) = 0.3, determine the smallest horizontal force P that must be applied to the lever to stop the wheel.

Blocks A and B have a mass of 7 kg and 10 kg, respectively. Using the coefficients of static friction indicated, determine the largest vertical force P which can be applied to the cord without causing motion.

The uniform bar AB is supported by a rope that passes over a frictionless pulley at C and a fixed peg at D. If the coefficient of static friction between the rope and the peg is \(\mu_{\mathrm{D}}\) = 0.3, determine the smallest distance x from the end of the bar at which a 20-N force may be placed and not cause the bar to move.

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 \(\mu_{\mathrm{s}}\) = 0.3

Blocks A and B weigh 50 lb and 30 lb, respectively. Using the coefficients of static friction indicated, determine the greatest weight of block D without causing motion.

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 \(\mu_{\mathrm{s}}\) = 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 angles u 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 \(\mu_{\mathrm{s}}\) = 0.1.

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 \(\mu_{\mathrm{s}}\) = 0.2.

The collar bearing uniformly supports an axial force of P = 5 kN. If the coefficient of static friction is \(\mu_{\mathrm{s}}\) = 0.3, determine the smallest torque M required to overcome friction.

The collar bearing uniformly supports an axial force of P = 8 kN. If a torque of M = 200 N \(\cdot\) m is applied to the shaft and causes it to rotate at constant velocity, determine the coefficient of kinetic friction at the surface of contact

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 \(\mu_{\mathrm{k}}\) = 0.3. Assume the brush exerts a uniform pressure on the floor.

The double-collar bearing is subjected to an axial force P = 4 kN. Assuming that collar A supports 0.75P and collar B supports 0.25P, both with a uniform distribution of pressure, determine the maximum frictional moment M that may be resisted by the bearing. Take \(\mu_{\mathrm{s}}\) = 0.2 for both collars

The double-collar bearing is subjected to an axial force P = 16 kN. Assuming that collar A supports 0.75P and collar B supports 0.25P, both with a uniform distribution of pressure, determine the smallest torque M that must be applied to overcome friction. Take \(\mu_{\mathrm{s}}\) = 0.2 for both collars.

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 m, determine the torque M required to overcome friction if the shaft supports an axial force P

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

The 4-in.-diameter shaft is held in the hole such that the normal pressure acting around the shaft varies linearly with its depth as shown. Determine the frictional torque that must be overcome to rotate the shaft. Take \(\mu_{\mathrm{s}}\) = 0.2.

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 \(\mu_{\mathrm{s}}\) = 0.4. Assume the bearing pressure between A and D to be uniform

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 \(\mu_{\mathrm{k}}\) = 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 \(\mu_{\mathrm{k}}\) = 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 pivot bearing is subjected to a parabolic pressure distribution at its surface of contact. If the coefficient of static friction is \(\mu_{\mathrm{k}}\), determine the torque M required to overcome friction and turn the shaft if it supports an axial force P.

A disk having an outer diameter of 120 mm fits loosely over a fixed shaft having a diameter of 30 mm. If the coefficient of static friction between the disk and the shaft is \(\mu_{\mathrm{s}}\) = 0.15 and the disk has a mass of 50 kg, determine the smallest vertical force F acting on the rim which must be applied to the disk to cause it to slip over the shaft.

The 4-lb pulley has a diameter of 1 ft and the axle has a diameter of 1 in. If the coefficient of kinetic friction between the axle and the pulley is \(\mu_{\mathrm{k}}\) = 0.20, determine the vertical force P on the rope required to lift the 20-lb block at constant velocity

Solve Prob. 8–120 if the force P is applied horizontally to the left.

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 \(\mu_{\mathrm{s}}\) = 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.

The uniform disk fits loosely over a fixed shaft having a diameter of 40 mm. If the coefficient of static friction between the disk and the shaft is \(\mu_{\mathrm{s}}\) = 0.15, determine the smallest vertical force P, acting on the rim, which must be applied to the disk to cause it to slip on the shaft. The disk has a mass of 20 kg

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 \(\mu_{\mathrm{k}}\) = 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 bell crank fits loosely into a 0.5-in-diameter pin. Determine the required force P which is just sufficient to rotate the bell crank clockwise. The coefficient of static friction between the pin and the bell crank is \(\mu_{\mathrm{s}}\) = 0.3.

The bell crank fits loosely into a 0.5-in-diameter pin. If P = 41 lb, the bell crank is then on the verge of rotating counterclockwise. Determine the coefficient of static friction between the pin and the bell crank.

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 handcart has wheels with a diameter of 6 in. If a crate having a weight of 1500 lb is placed on the cart, determine the force P that must be applied to the handle to overcome the rolling resistance. The coefficient of rolling resistance is 0.04 in. Neglect the weight 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 \(P=\left[\dot{W}\left(a_{A}+a_{B}\right)\right] / 2 r\) is required to move the load and thereby roll the cylinder forward. Neglect the weight of the cylinder

The 1.4-Mg machine is to be moved over a level surface using a series of rollers for which the coefficient of rolling resistance is 0.5 mm at the ground and 0.2 mm at the bottom surface of the machine. Determine the appropriate diameter of the rollers so that the machine can be pushed forward with a horizontal force of P = 250 N. Hint: Use the result of Prob. 8–131