The wheel of radius R rolls without slipping, and its center O has an acceleration . A

The circular disk of radius rotates about a fixed axis through point O with the angular properties rad/s and rad/s2 with directions as shown in the figure. Determine the instantaneous values of the velocity and acceleration of point A. Problem 5/1

The triangular plate rotates about a fixed axis through point O with the angular properties indicated. Determine the instantaneous velocity and acceleration of point A. Take all given variables to be positive. Problem 5/2

The body is formed of slender rod and rotates about a fixed axis through point O with the indicated angular properties. If rad/s and rad/s2 , determine the instantaneous velocity and acceleration of point A. Problem 5/3

A torque applied to a flywheel causes it to accelerate uniformly from a speed of 200 rev/min to a speed of 800 rev/min in 4 seconds. Determine the number of revolutions N through which the wheel turns during this interval. (Suggestion: Use revolutions and minutes for units in your calculations.)

The drive mechanism imparts to the semicircular plate simple harmonic motion of the form , where is the amplitude of the oscillation and is its circular frequency. Determine the amplitudes of the angular velocity and angular acceleration and state where in the motion cycle these maxima occur. Note that this motion is not that of a freely pivoted and undriven body undergoing arbitrarily large-amplitude angular motion. Problem 5/5 O

The mass center G of the car has a velocity of 40 mi/hr at position A and 1.52 seconds later at B has a velocity of 50 mi/hr. The radius of curvature of the road at B is 180 ft. Calculate the angular velocity of the car at B and the average angular velocity of the car between A and B.

The rectangular plate is rotating about its corner axis through O with a constant angular velocity rad/s. Determine the magnitudes of the velocity v and acceleration a of the corner A by (a) using the scalar relations and (b) using the vector relations.

If the rectangular plate of Prob. 5/7 starts from rest and point B has an initial acceleration of 5.5 m/s2 , determine the distance b if the plate reaches an angular speed of 300 rev/min in 2 seconds with a constant angular acceleration.

A shaft is accelerated from rest at a constant rate to a speed of 3600 rev/min and then is immediately decelerated to rest at a constant rate within a total time of 10 seconds. How many revolutions N has the shaft turned during this interval?

The bent flat bar rotates about a fixed axis through point O. At the instant depicted, its angular properties are rad/s and with directions as indicated in the figure. Determine the instantaneous velocity and acceleration of point A. Problem 5/10

The angular acceleration of a body which is rotating about a fixed axis is given by , where the constant (no units). Determine the angular displacement and time elapsed when the angular velocity has been reduced to one-third its initial value rad/s.

The angular position of a radial line in a rotating disk is given by the clockwise angle where is in radians and t is in seconds. Calculate the angular displacement of the disk during the interval in which its angular acceleration increases from 42 rad/s2 to 66 rad/s2

In order to test an intentionally weak adhesive, the bottom of the small 0.3-kg block is coated with adhesive and then the block is pressed onto the turntable with a known force. The turntable starts from rest at time and uniformly accelerates with . If the adhesive fails at exactly determine the ultimate shear force which the adhesive supports. What is the angular displacement of the turntable at the time of failure?

The plate OAB forms an equilateral triangle which rotates counterclockwise with increasing speed about point O. If the normal and tangential components of acceleration of the centroid C at a certain instant are 80 m/s2 and 30 m/s2 , respectively, determine the values of and at this same instant. The angle is the angle between line AB and the fixed horizontal axis. Problem 5/14 A B C O 150 mm 150 mm 150 mm

Experimental data for a rotating control element reveal the plotted relation between angular velocity and the angular coordinate as shown. Approximate the angular acceleration of the element when Problem 5/15

The rotating arm starts from rest and acquires a rotational speed rev/min in 2 seconds with constant angular acceleration. Find the time t after starting before the acceleration vector of end P makes an angle of with the arm OP. Problem 5/16 P N O

The belt-driven pulley and attached disk are rotating with increasing angular velocity. At a certain instant the speed v of the belt is 1.5 m/s, and the total acceleration of point A is 75 m/s2 . For this instant determine (a) the angular acceleration of the pulley and disk, (b) the total acceleration of point B, and (c) the acceleration of point C on the belt. Problem 5/17

Magnetic tape is being fed over and around the light pulleys mounted in a computer. If the speed v of the tape is constant and if the magnitude of the acceleration of point A on the tape is 4/3 times that of point B, calculate the radius r of the smaller pulley. Problem 5/18

The circular disk rotates about its center O. For the instant represented, the velocity of A is in./sec and the tangential acceleration of B is in./sec2 . Write the vector expressions for the angular velocity and angular acceleration of the disk. Use these results to write the vector expression for the acceleration of point C.

Point A of the circular disk is at the angular position at time The disk has angular velocity at and subsequently experiences a constant angular acceleration Determine the velocity and acceleration of point A in terms of fixed i and j unit vectors at time Problem 5/20

Repeat Prob. 5/20, except now the angular acceleration of the disk is given by where t is in seconds and is in radians per second squared. Determine the velocity and acceleration of point A in terms of fixed i and j unit vectors at time s.

Repeat Prob. 5/20, except now the angular acceleration of the disk is given by , where is in radians per second and is in radians per second squared. Determine the velocity and acceleration of point A in terms of fixed i and j unit vectors at time s.

The disk of Prob. 5/20 is at the angular position at time . Its angular velocity at is rad/s, and then it experiences an angular acceleration given by , where is in radians and is in radians per second squared. Determine the angular position of point A at time s. t

During its final spin cycle, a front-loading washing machine has a spin rate of 1200 rev/min. Once power is removed, the drum is observed to uniformly decelerate to rest in 25 s. Determine the number of revolutions made during this period as well as the number of revolutions made during the first half of it. Problem 5/24

The solid cylinder rotates about its z-axis. At the instant represented, point P on the rim has a velocity whose x-component is ft/sec, and . Determine the angular velocity of line AB on the face of the cylinder. Does the element line BC have an angular velocity? Problem 5/25

The two V-belt pulleys form an integral unit and rotate about the fixed axis at O. At a certain instant, point A on the belt of the smaller pulley has a velocity , and point B on the belt of the larger pulley has an acceleration as shown. For this instant determine the magnitude of the acceleration of point C and sketch the vector in your solution. aC

A clockwise variable torque is applied to a flywheel at time causing its clockwise angular acceleration to decrease linearly with angular displacement during 20 revolutions of the wheel as shown. If the clockwise speed of the flywheel was 300 rev/min at , determine its speed N after turning the 20 revolutions. (Suggestion: Use units of revolutions instead of radians.) Problem 5/27

The design characteristics of a gear-reduction unit are under review. Gear B is rotating clockwise with a speed of 300 rev/min when a torque is applied to gear A at time s to give gear A a counterclockwise acceleration which varies with time for a duration of 4 seconds as shown. Determine the speed of gear B when Problem 5/28

The fixed hydraulic cylinder C imparts a constant upward velocity v to the collar B, which slips freely on rod OA. Determine the resulting angular velocity in terms of v, the displacement s of point B, and the fixed distance d. Problem 5/29

Point A is given a constant acceleration a to the right starting from rest with x essentially zero. Determine the angular velocity of link AB in terms of x and a. Problem 5/30

The telephone-cable reel is rolled down the incline by the cable leading from the upper drum and wrapped around the inner hub of the reel. If the upper drum is turned at the constant rate calculate the time required for the center of the reel to move 100 ft along the incline. No slipping occurs. Problem 5/31

The small vehicle rides on rails and is driven by the 400-mm-diameter friction wheel turned by an electric motor. Determine the speed v of the vehicle if the friction-drive wheel is rotating at a speed of 300 rev/min and if no slipping occurs. Problem 5/32

The Scotch-yoke mechanism converts rotational motion of the disk to oscillatory translation of the shaft. For given values of , and d, determine the velocity and acceleration of point P of the shaft. Problem 5/33

Slider A moves in the horizontal slot with a constant speed v for a short interval of motion. Determine the angular velocity of bar AB in terms of the displacement Problem 5/34

The cables at A and B are wrapped securely around the rims and the hub of the integral pulley as shown. If the cables at A and B are given upward velocities of 3 ft/sec and 4 ft/sec, respectively, calculate the velocity of the center O and the angular velocity of the pulley. Problem 5/35

The wheel of radius r rolls without slipping, and its center O has a constant velocity to the right. Determine expressions for the magnitudes of the velocity v and acceleration a of point A on the rim by differentiating its x- and y-coordinates. Represent your results graphically as vectors on your sketch and show that v is the vector sum of two vectors, each of which has a magnitude Problem 5/36 y x A r O

Determine the acceleration of the shaft B for if the crank OA has an angular acceleration and an angular velocity at this position. The spring maintains contact between the roller and the surface of the plunger. Problem 5/37

The collar C moves to the left on the fixed guide with speed v. Determine the angular velocity as a function of v, the collar position s, and the height h. Problem 5/38

The linear actuator is designed for rapid horizontal velocity v of jaw C for a slow change in the distance between A and B. If the hydraulic cylinder decreases this distance at the rate u, determine the horizontal velocity of jaw C in terms of the angle Problem 5/39

The telephone-cable reel rolls without slipping on the horizontal surface. If point A on the cable has a velocity to the right, compute the velocity of the center O and the angular velocity of the reel. (Be careful not to make the mistake of assuming that the reel rolls to the left.) Problem 5/4

As end A of the slender bar is pulled to the right with the velocity v, the bar slides on the surface of the fixed half-cylinder. Determine the angular velocity of the bar in terms of x. Problem 5/41 A r x v B

Calculate the angular velocity of the slender bar AB as a function of the distance x and the constant angular velocity of the drum. Problem 5/42

The circular cam is mounted eccentrically about its fixed bearing at O and turns counterclockwise at the constant angular velocity The cam causes the fork A and attached control rod to oscillate in the horizontal x-direction. Write expressions for the velocity and acceleration of the control rod in terms of the angle measured from the vertical. The contact surfaces of the fork are vertical.

Rotation of the lever OA is controlled by the motion of the contacting circular disk whose center is given a horizontal velocity v. Determine the expression for the angular velocity of the lever OA in terms of x. Problem 5/44

Motion of the sliders B and C in the horizontal guide is controlled by the vertical motion of the slider A. If A is given an upward velocity determine as a function of the magnitude v of the equal and opposite velocities which B and C have as they move toward one another. Problem 5/45

Derive an expression for the upward velocity v of the car hoist in terms of . The piston rod of the hydraulic cylinder is extending at the rate . Problem 5/46

The cable from drum A turns the double wheel B, which rolls on its hubs without slipping. Determine the angular velocity and angular acceleration of drum C for the instant when the angular velocity and angular acceleration of A are and respectively, both in the counterclockwise direction.

The flywheel turns clockwise with a constant speed of The connecting link AB slides through the pivoted collar at C. Calculate the angular velocity of AB for the instant when Problem

For the instant represented when the piston rod of the hydraulic cylinder C imparts a vertical motion to the pin B consisting of and For this instant determine the angular velocity and the angular acceleration of link OA. Members OA and AB make equal angles with the horizontal at this instant. Problem 5/4

Link OA has an angular velocity as it passes the position shown. Determine the corresponding angular velocity of the slotted link CB. Solve by considering the relation between the infinitesimal displacements involved. Problem 5/50

Show that the expressions and hold for the motion of the center O of the wheel which rolls on the concave or convex circular arc, where and are the absolute angular velocity and acceleration, respectively, of the wheel. (Hint: Follow the example of Sample Problem 5/4 and allow the wheel to roll a small distance. Be very careful to identify the correct absolute angle through which the wheel turns in each case in determining its angular velocity and angular acceleration.) Problem 5/51

Film passes through the guide rollers shown and is being wound onto the reel, which is turned at a constant angular velocity . Determine the acceleration of the film as it enters the rollers. The thickness of the film is t, and s is sufficiently large so that the change in the angle made by the film with the horizontal is negligible.

Angular oscillation of the slotted link is achieved by the crank OA, which rotates clockwise at the steady speed Determine an expression for the angular velocity of the slotted link in terms of . Problem 5/53

Link OA revolves counterclockwise with an angular velocity of Link AB slides through the pivoted collar at C. Determine the angular velocity of AB when Problem 5/54

The piston rod of the hydraulic cylinder gives point B a velocity as shown. Determine the magnitude of the velocity of point C in terms of . Problem 5/55

The Geneva wheel is a mechanism for producing intermittent rotation. Pin P in the integral unit of wheel A and locking plate B engages the radial slots in wheel C, thus turning wheel C one-fourth of a revolution for each revolution of the pin. At the engagement position shown, For a constant clockwise angular velocity of wheel A, determine the corresponding counterclockwise angular velocity of wheel C for (Note that the motion during engagement is governed by the geometry of triangle with changing .) Problem 5/56 200 mm P A C B O1 O2

One of the most common mechanisms is the slidercrank. Express the angular velocity and angular acceleration of the connecting rod AB in terms of the crank angle for a given constant crank speed Take and to be positive counterclockwise. Problem 5/57

The rod AB slides through the pivoted collar as end A moves along the slot. If A starts from rest at and moves to the right with a constant acceleration of , calculate the angular acceleration of AB at the

Bar AB moves on the horizontal surface. Its mass center has a velocity directed parallel to the y-axis and the bar has a counterclockwise (as seen from above) angular velocity Determine the velocity of point B. Problem 5/59

The cart has a velocity of 4 ft/sec to the right. Determine the angular speed N of the wheel so that point A on the top of the rim has a velocity (a) equal to 4 ft/sec to the left, (b) equal to zero, and (c) equal to 8 ft/sec to the right. Problem 5/60

The speed of the center of the earth as it orbits the sun is and the absolute angular velocity of the earth about its north-south spin axis is Use the value for the radius of the earth and determine the velocities of points A, B, C, and D, all of which are on the equator. The inclination of the axis of the earth is neglected. Problem 5/61

A control element in a special-purpose mechanism undergoes motion in the plane of the figure. If the velocity of B with respect to A has a magnitude of 0.926 m/s at a certain instant, what is the corresponding magnitude of the velocity of C with respect to D? Problem 5/62

End A of the 24-in. link has a velocity of 10 ft/sec in the direction shown. At the same instant end B has a velocity whose magnitude is 12 ft/sec as indicated. Find the angular velocity of the link in two ways. Problem 5/63

The circular disk of radius 0.2 m is released very near the horizontal surface with a velocity of its center to the right and a clockwise angular velocity Determine the velocities of points A and P of the disk. Describe the motion upon contact with the ground. Problem 5/64

65 For the instant represented the curved link has a counterclockwise angular velocity of 4 rad/s, and the roller at B has a velocity of 40 mm/s along the constraining surface as shown. Determine the magnitude of the velocity of A. Problem 5/65

Determine the angular velocity of the telescoping link AB for the position shown where the driving links have the angular velocities indicated. Problem 5/66

The ends of bar AB are confined to the circular slot. By the method of this article, determine the angular velocity of the bar if the velocity of end A is as shown. Problem 5/67

The two pulleys are riveted together to form a single rigid unit, and each of the two cables is securely wrapped around its respective pulley. If point A on the hoisting cable has a velocity determine the magnitudes of the velocity of point O and the velocity of point B on the larger pulley for the position shown. Problem 5/68

The right-angle link AB has a clockwise angular velocity of at the instant when Express the velocity of A with respect to B in vector notation for this instant. Problem 5/69 12 A B 5 3 rad/sec y

The magnitude of the absolute velocity of point A on the automobile tire is when A is in the position shown. What are the corresponding velocity of the car and the angular velocity of the wheel? (The wheel rolls without slipping.) Problem 5/70

For the instant represented point B crosses the horizontal axis through point O with a downward velocity Determine the corresponding value of the angular velocity of link OA. Problem 5/71

The circular disk rolls without slipping with a clockwise angular velocity For the instant represented, write the vector expressions for the velocity of A with respect to B and for the velocity of P. Problem 5/72 B A C

At the instant represented, the velocity of point A of the 1.2-m bar is to the right. Determine the speed of point B and the angular velocity of the bar. The diameter of the small end wheels may be neglected. Problem 5/73

For an interval of its motion the piston rod of the hydraulic cylinder has a velocity as shown. At a certain instant For this instant determine the angular velocity of link BC. Problem 5/74

Each of the sliding bars A and B engages its respective rim of the two riveted wheels without slipping. Determine the magnitude of the velocity of point P for the position shown. Problem 5/75

Determine the angular velocity of link AB and the velocity of collar B for the instant represented. Assume the quantities and r to be known. Problem 5/76

Determine the angular velocity of link AB and the velocity of collar B for the instant represented. Assume the quantities and r to be known. Problem 5/77

Motion of the threaded collars A and B is controlled by the rotation of their respective lead screws. If A has a velocity to the right of and B has a velocity to the left of when in., determine the angular velocity of ACD at this instant. Problem 5/78

At the instant represented the triangular plate ABD has a clockwise angular velocity of For this instant determine the angular velocity of link BC. Problem 5/79

For the instant represented, crank OB has a clockwise angular velocity and is passing the horizontal position. Determine the corresponding velocity of the guide roller A in the 20 slot and the velocity of point C midway between A and B. Problem 5/80 C O 20 20

The ends of the 0.4-m slender bar remain in contact with their respective support surfaces. If end B has a velocity in the direction shown, determine the angular velocity of the bar and the velocity of end A. Problem 5/81

End A of the link has a downward velocity of during an interval of its motion. For the position where determine the angular velocity of AB and the velocity of the midpoint G of the link. Solve the relative-velocity equations, first, using the geometry of the vector polygon and, second, using vector algebra. Problem 5/82

Horizontal motion of the piston rod of the hydraulic cylinder controls the rotation of link OB about O. For the instant represented, and OB is horizontal. Determine the angular velocity of OB for this instant. Problem 5/83

The flywheel turns clockwise with a constant speed of and the connecting rod AB slides through the pivoted collar at C. For the position determine the angular velocity of AB by using the relative-velocity relations. (Suggestion: Choose a point D on AB coincident with C as a reference point whose direction of velocity is known.) Problem 5/84

Determine the velocity of point D which will produce a counterclockwise angular velocity of for link AB in the position shown for the four-bar linkage. Problem 5/85 75 mm A O C B D 150 mm 100 mm 100 mm

The elements of a switching device are shown. If the vertical control rod has a downward velocity v of when and if roller A is in continuous contact with the horizontal surface, determine the magnitude of the velocity of C for this instant. Problem 5/86

The Geneva mechanism of Prob. 5/56 is shown again here. By relative-motion principles determine the angular velocity of wheel C for Wheel A has a constant clockwise angular velocity Problem 5/87

A four-bar linkage is shown in the figure (the ground link OC is considered the fourth bar). If the drive link OA has a counterclockwise angular velocity determine the angular velocities of links AB and BC. Problem 5/88

The elements of the mechanism for deployment of a spacecraft magnetometer boom are shown. Determine the angular velocity of the boom when the driving link OB crosses the y-axis with an angular velocity if at this instant. Problem 5/89

Ends A and C of the connected links are controlled by the vertical motion of the piston rods of the hydraulic cylinders. For a short interval of motion, A has an upward velocity of and C has a downward velocity of Determine the velocity of B for the instant when Problem 5/90

The slender bar is moving in general plane motion with the indicated linear and angular properties. Locate the instantaneous center of zero velocity and determine the velocities of points A and B. Problem 5/91

The slender bar is moving in general plane motion with the indicated linear and angular properties. Locate the instantaneous center of zero velocity and determine the velocities of points A and B. Problem 5/92

The figure for Prob. 5/83 is repeated here. Solve for the angular velocity of OB by the method of this article. Problem 5/93

The circular disk of Prob. 5/64 is repeated here. If the disk is released very near the horizontal surface with and locate the instantaneous center of rotation and determine the velocities of points A and P of the disk. Problem 5/94

For the instant represented, when crank OA passes the horizontal position, determine the velocity of the center G of link AB by the method of this article. Problem 5/95

The bar AB has a clockwise angular velocity of Construct and determine the vector velocity of each end if the instantaneous center of zero velocity is (a) at and (b) at . Problem 5/96

The bar of Prob. 5/81 is repeated here. By the method of this article, determine the velocity of end A. Both ends remain in contact with their respective support surfaces. Problem 5/97

A car mechanic walks two wheel/tire units across a horizontal floor as shown. He walks with constant speed v and keeps the tires in the configuration shown with the same position relative to his body. If there is no slipping at any interface, determine (a) the angular velocity of the lower tire, (b) the angular velocity of the upper tire, and (c) the velocities of points A, B, C, and D. The radius of both tires is r. Problem 5/98 v A

The linkage of Prob. 5/80 is repeated here. At the instant represented, crank OB has a clockwise angular velocity and is passing the horizontal position. By the method of this article, determine the corresponding velocity of the guide roller A in the 20 slot and the velocity of point C midway between A and B. Problem 5/99

Motion of the bar is controlled by the constrained paths of A and B. If the angular velocity of the bar is counterclockwise as the position is passed, determine the speeds of points A and P. Problem 5/100

Motion of the rectangular plate P is controlled by the two links which cross without touching. For the instant represented where the links are perpendicular to each other, the plate has a counterclockwise angular velocity Determine the corresponding angular velocities of the two links. Problem 5/101

The mechanism of Prob. 5/34 is repeated here. At the instant when the velocity of the slider at A is to the right. Determine the corresponding velocity of slider B and the angular velocity of bar AB if Problem 5/102 xA B A v

The mechanism of Prob. 5/74 is repeated here. For an interval of its motion the piston rod of the hydraulic cylinder has a velocity as shown. At a certain instant By the method of this article, determine the angular velocity of link BC. Problem 5/103

The mechanism of Prob. 5/76 is repeated here. By the method of this article, determine the angular velocity of link AB and the velocity of collar B for the instant shown. Assume the quantities and r to be known. Problem 5/104

The mechanism of Prob. 5/77 is repeated here. By the method of this article, determine the angular velocity of link AB and the velocity of collar B for the instant depicted. Assume the quantities and r to be known. Problem 5/105

The rectangular body B is pivoted to the crank OA at A and is supported by the wheel at D. If OA has a counterclockwise angular velocity of determine the velocity of point E and the angular velocity of body B when the crank OA passes the vertical position shown. Problem 5/106

The sliding rails A and B engage the rims of the double wheel without slipping. For the specified velocities of A and B, determine the angular velocity of the wheel and the magnitude of the velocity of point P. Problem 5/107

Horizontal oscillation of the spring-loaded plunger E is controlled by varying the air pressure in the horizontal pneumatic cylinder F. If the plunger has a velocity of to the right when determine the downward velocity of roller D in the vertical guide and find the angular velocity of ABD for this position. Problem 5/108 100 mm 200

The rear driving wheel of a car has a diameter of 26 in. and has an angular speed N of on an icy road. If the instantaneous center of zero velocity is 4 in. above the point of contact of the tire with the road, determine the velocity v of the car and the slipping velocity of the tire on the ice. Problem 5/10

The elements of the mechanism for deployment of a spacecraft magnetometer boom are repeated here from Prob. 5/89. By the method of this article, determine the angular velocity of the boom when the driving link OB crosses the y-axis with an angular velocity if at this instant Problem 5/110

The mechanism of Prob. 5/55 is repeated here. By the method of this article determine the expression for the magnitude of the velocity of point C in terms of the velocity of the piston rod and the angle . Problem 5/111

Link OA has a counterclockwise angular velocity during an interval of its motion. Determine the angular velocity of link AB and of sector BD for at which instant AB is horizontal and BD is vertical. Problem 5/112

The mechanism of Prob. 5/84 is repeated here. The flywheel turns clockwise with a constant speed of and the connecting rod AB slides through the pivoted collar at C. For the position determine the angular velocity of AB by the method of this article. Problem 5/113

The hydraulic cylinder produces a limited horizontal motion of point A. If when determine the magnitude of the velocity of D and the angular velocity of ABD for this position. Problem 5/114

The flexible band F is attached at E to the rotating sector and leads over the guide pulley. Determine the angular velocities of AD and BD for the position shown if the band has a velocity of Problem 5/115

Motion of the roller A against its restraining spring is controlled by the downward motion of the plunger E. For an interval of motion the velocity of E is Determine the velocity of A when becomes 90.

In the design of this mechanism, upward motion of the plunger G controls the motion of a control rod attached at A. Point B of link AH is confined to move with the sliding collar on the fixed vertical shaft ED. If G has a velocity for a short interval, determine the velocity of A for the position . Problem 5/117

Determine the angular velocity of the ram head AE of the rock crusher in the position for which . The crank OB has an angular speed of When B is at the bottom of its circle, D and E are on a horizontal line through F, and lines BD and AE are vertical. The dimensions are , and Carefully construct the configuration graphically, and use the method of this article. Problem 5/118

The large roller bearing rolls to the left on its outer race with a velocity of its center O of At the same time the central shaft and inner race rotate counterclockwise with an angular speed of Determine the angular velocity of each of the rollers.

The shaft at O drives the arm OA at a clockwise speed of about the fixed bearing at O. Use the method of the instantaneous center of zero velocity to determine the rotational speed of gear B (gear teeth not shown) if (a) ring gear D is fixed and (b) ring gear D rotates counterclockwise about O with an angular speed of Problem 5/120

The center O of the wheel is mounted on the sliding block, which has an acceleration to the right. At the instant when , and . For this instant determine the magnitudes of the accelerations of points A and B. Problem 5/121

The 9-ft steel beam is being hoisted from its horizontal position by the two cables attached at A and B. If the initial angular accelerations are and , determine the initial values of (a) the angular acceleration of the beam, (b) the acceleration of point C, and (c) the distance d from A to the point on the centerline of the beam which has zero acceleration. Problem 5/122

The two rotor blades of 800-mm radius rotate counterclockwise with a constant angular velocity about the shaft at O mounted in the sliding block. The acceleration of the block is . Determine the magnitude of the acceleration of the tip A of the blade when (a) (b) , and (c) . Does the velocity of O or the sense of enter into the calculation?

Refer to the rotor blades and sliding bearing block of Prob. 5/123 where . If and when , find the acceleration of point A for this instant.

The wheel of radius R rolls without slipping, and its center O has an acceleration . A point P on the wheel is a distance r from O. For given values of , R, and r, determine the angle and the velocity of the wheel for which P has no acceleration in this position. Problem 5/125

The circular disk rolls to the left without slipping. If , determine the velocity and acceleration of the center O of the disk. Problem 5/126 B A 150 mm 200 mm 150 mm

The bar of Prob. 5/81 is repeated here. The ends of the 0.4-m bar remain in contact with their respective support surfaces. End B has a velocity of and an acceleration of in the directions shown. Determine the angular acceleration of the bar and the acceleration of end A. Problem 5/127

Determine the acceleration of point B on the equator of the earth, repeated here from Prob. 5/61. Use the data given with that problem and assume that the earths orbital path is circular, consulting Table D/2 as necessary. Consider the center of the sun fixed and neglect the tilt of the axis of the earth. Problem 5/128 x A v B Sunlight C D y N

A car with tires of 600-mm diameter accelerates at a constant rate from rest to a velocity of 60 km/h in a distance of 40 m. Determine the magnitude of the acceleration of a point A on the top of the wheel as the car reaches a speed of 10 km/h.

A car has a forward acceleration without slipping its 24-in.-diameter tires. Determine the velocity v of the car when a point P on the tire in the position shown will have zero horizontal component of acceleration. Problem 5/130

Determine the angular acceleration of AB for the position shown if link OB has a constant angular velocity Problem 5/131 r r A O B r

Determine the angular acceleration of link AB and the linear acceleration of A for if and at that position. Carry out your solution using vector notation. Problem 5/132

The end rollers of bar AB are constrained to the slot shown. If roller A has a downward velocity of 1.2 m/s and this speed is constant over a small motion interval, determine the tangential acceleration of roller B as it passes the topmost position. The value of R is 0.5 m. Problem 5/133

The load L is lowered by the two pulleys which are fastened together and rotate as a single unit. For the instant represented, drum A has a counterclockwise angular velocity of which is decreasing by each second. Simultaneously, drum B has a clockwise angular velocity of which is increasing by each second. Calculate the accelerations of points C and D and the load L. Problem 5/134

The mechanism of Prob. 5/76 is repeated here. The angular velocity of the disk is constant. For the instant represented, determine the angular acceleration of link AB and the acceleration of collar B. Assume the quantities and r to be known. Problem 5/135

Crank OA oscillates between the dashed positions shown and causes small angular motion of crank BC through the connecting link AB. When OA crosses the horizontal position with AB horizontal and BC vertical, it has an angular velocity and zero angular acceleration. Determine the angular acceleration of BC for this position. Problem 5/136

The shaft of the wheel unit rolls without slipping on the fixed horizontal surface. If the velocity and acceleration of point O are to the right and to the left, respectively, determine the accelerations of points A and D. Problem 5/137

The hydraulic cylinder imparts motion to point B which causes link OA to rotate. For the instant shown where OA is vertical and AB is horizontal, the velocity of pin B is and is increasing at the rate of For this position determine the angular acceleration of OA.

The velocity of roller A is to the right as shown, and this velocity is momentarily decreasing at a rate of Determine the corresponding value of the angular acceleration of bar AB as well as the tangential acceleration of roller B along the circular guide. The value of R is 0.6 m. Problem 5/139

The bar AB from Prob. 5/73 is repeated here. If the velocity of point A is to the right and is constant for an interval including the position shown, determine the tangential acceleration of point B along its path and the angular acceleration of the bar. Problem 5/140 1.2 m

The center O of the wooden spool is moving vertically downward with a speed and this speed is increasing at the rate of Determine the accelerations of points A, P, and B. Problem 5/141

Link OA has a constant counterclockwise angular velocity during a short interval of its motion. For the position shown determine the angular accelerations of AB and BC. Problem 5/142

The linkage of Prob. 5/74 is shown again here. For the instant when the hydraulic cylinder gives A a velocity which is increasing by each second. For this instant determine the angular acceleration of link BC.

The sliding collar moves up and down the shaft, causing an oscillation of crank OB. If the velocity of A is not changing as it passes the null position where AB is horizontal and OB is vertical, determine the angular acceleration of OB in that position. Problem 5/144

For the linkage shown, if and is constant when the two links become perpendicular to one another, determine the angular acceleration of CB for this position. Problem 5/145

The mechanism of Prob. 5/75 is repeated here. Each of the sliding bars A and B engages its respective rim of the two riveted wheels without slipping. If, in addition to the information shown, bar A has an acceleration of to the right and there is no acceleration of bar B, calculate the magnitude of the acceleration of P for the instant depicted. Problem 5/146

The four-bar linkage of Prob. 5/88 is repeated here. If the angular velocity and angular acceleration of drive link OA are and respectively, both counterclockwise, determine the angular accelerations of bars AB and BC for the instant represented. Problem 5/147

The elements of a simplified clam-shell bucket for a dredge are shown. With the block at O considered fixed and with the constant velocity of the control cable at C equal to determine the angular acceleration of the right-hand bucket jaw when 45 as the bucket jaws are closing

The revolving crank ED and connecting link CD cause the rigid frame ABO to oscillate about O. For the instant represented ED and CD are both perpendicular to FO, and the crank ED has an angular velocity of and an angular acceleration of both counterclockwise. For this instant determine the acceleration of point A with respect to point B. Problem 5/149

If link AB of the four-bar linkage has a constant counterclockwise angular velocity of during an interval which includes the instant represented, determine the angular acceleration of AO and the acceleration of point D. Express your results in vector notation. Problem 5/150

The crank OA of the offset slider-crank mechanism rotates with a constant clockwise angular velocity Determine the angular acceleration of link AB and the acceleration of B for the depicted position. Problem 5/151

For a short interval of motion, link OA has a constant angular velocity Determine the angular acceleration of link AB for the instant when OA is parallel to the horizontal axis through B. Problem 5/152

The elements of a power hacksaw are shown in the figure. The saw blade is mounted in a frame which slides along the horizontal guide. If the motor turns the flywheel at a constant counterclockwise speed of determine the acceleration of the blade for the position where and find the corresponding angular acceleration of the link AB. Problem 5/153

The mechanism of Prob. 5/115 is repeated here where the flexible band F attached to the sector at E is given a constant velocity of as shown. For the instant when BD is perpendicular to OA, determine the angular acceleration of BD. Problem 5/154

An oil pumping rig is shown in the figure. The flexible pump rod D is fastened to the sector at E and is always vertical as it enters the fitting below D. The link AB causes the beam BCE to oscillate as the weighted crank OA revolves. If OA has a constant clockwise speed of 1 rev every 3 s, determine the acceleration of the pump rod D when the beam and the crank OA are both in the horizontal position shown.

A mechanism for pushing small boxes from an assembly line onto a conveyor belt is shown with arm OD and crank CB in their vertical positions. For the configuration shown, crank CB has a constant clockwise angular velocity of Determine the acceleration of E. Problem 5/156

The disk rotates with angular speed The small ball A is moving along the radial slot with speed relative to the disk. Determine the absolute velocity of the ball and state the angle between this velocity vector and the positive x-axis. Problem 5/157

In addition to the conditions stated in the previous problem, the ball speed u (relative to the disk) is increasing at the rate of and the angular rate of the disk is decreasing at the rate of Determine the Coriolis acceleration relative to the disk-fixed Bxy coordinate system. Also determine the absolute acceleration of ball A and the angle between this acceleration vector and the positive x-axis.

The disk rotates about a fixed axis through O with angular velocity and angular acceleration at the instant represented, in the directions shown. The slider A moves in the straight slot. Determine the absolute velocity and acceleration of A for the same instant, when in., and y 30 in./sec2 y 24 in./sec, .

The disk rotates about a fixed axis through O with angular velocity and angular acceleration in the directions shown at a certain instant. The small sphere A moves in the circular slot, and at the same instant, and Determine the absolute velocity and acceleration of A at this instant. Problem 5/160 y

The slotted wheel rolls to the right without slipping, with a constant speed of its center O. Simultaneously, motion of the sliding block A is controlled by a mechanism not shown so that with Determine the magnitude of the acceleration of A for the instant when and . Problem 5/161

The disk rolls without slipping on the horizontal surface, and at the instant represented, the center O has the velocity and acceleration shown in the figure. For this instant, the particle A has the indicated speed u and time rate of change of speed , both relative to the disk. Determine the absolute velocity and acceleration of particle A. Problem 5/162

An experimental vehicle A travels with constant speed v relative to the earth along a northsouth track. Determine the Coriolis acceleration as a function of the latitude . Assume an earth-fixed rotating frame Bxyz and a spherical earth. If the vehicle speed is , determine the magnitude of the Coriolis acceleration at (a) the equator and (b) the north pole.

A stationary pole A is viewed by an observer P who is sitting on a small merry-go-round which rotates about a fixed vertical axis at B with a constant angular velocity as shown. Determine the apparent velocity of A as seen by the observer P. Does this velocity depend on the location of the observer on the merry-go-round? Problem 5/164

The small collar A is sliding on the bent bar with speed u relative to the bar as shown. Simultaneously, the bar is rotating with angular velocity about the fixed pivot B. Take the x-y axes to be fixed to the bar and determine the Coriolis acceleration of the slider for the instant represented. Interpret your result. Problem 5/165

The fire truck is moving forward at a speed of 35 mi/hr and is decelerating at the rate of Simultaneously, the ladder is being raised and extended. At the instant considered the angle is and is increasing at the constant rate of 10 deg/sec. Also at this instant the extension b of the ladder is 5 ft, with and For this instant determine the acceleration of the end A of the ladder (a) with respect to the truck and (b) with respect to the ground. Problem 5/166

For an alternative solution to Prob. assign r- coordinates with origin at B as shown. Then make use of the polar-coordinate relations for the acceleration of A relative to B. The r- and components of the absolute acceleration should coincide with the components along and normal to the ladder which would be found in Prob. 5/166. Problem 5/167

Aircraft B has a constant speed of at the bottom of a circular loop of 400-m radius. Aircraft A flying horizontally in the plane of the loop passes 100 m directly under B at a constant speed of . With coordinate axes attached to B as shown, determine the acceleration which A appears to have to the pilot of B for this instant. Problem 5/168

Bar OA has a counterclockwise angular velocity Rod BC slides freely through the pivoted collar attached to OA. Determine the angular velocity of rod BC and the velocity of collar A relative to rod BC. Problem 5/169

A smooth bowling alley is oriented northsouth as shown. A ball A is released with speed v along the lane as shown. Because of the Coriolis effect, it will deflect a distance as shown. Develop a general expression for . The bowling alley is located at a latitude in the northern hemisphere. Evaluate your expression for the conditions and Should bowlers prefer eastwest alleys? State any assumptions. Problem 5/170 Not to scale

Under the action of its stern and starboard bow thrusters, the cruise ship has the velocity of its mass center B and angular velocity about a vertical axis. The velocity of B is constant, but the angular rate is decreasing at Person A is stationary on the dock. What velocity and acceleration of A are observed by a passenger fixed to and rotating with the ship? Treat the problem as two-dimensional. Problem 5/171

All conditions of the previous problem remain, except now person A is running to the right with a constant speed (with his instantaneous location still as indicated in the figure). Determine the velocity and acceleration which A appears to have relative to a passenger fixed to and rotating with the ship

Two boys A and B are sitting on opposite sides of a horizontal turntable which rotates at a constant counterclockwise angular velocity as seen from above. Boy A throws a ball toward B by giving it a horizontal velocity u relative to the turntable toward B. Assume that the ball has no horizontal acceleration once released and write an expression for the magnitude of the acceleration which B would observe the ball to have in the plane of the turntable just after it is thrown. Sketch the path of the ball on the turntable as observed by B. Problem 5/173

Car B turns onto the circular off-ramp with a speed v. Car A, traveling with the same speed v, continues in a straight line. Prove that the velocity which A appears to have to an observer riding in and turning with car B is zero when car A passes the position shown regardless of the angle Problem 5/174

For the conditions and conclusion of Prob. 5/174, show that the acceleration which car A appears to have to an observer in and turning with car B is equal to in the direction normal to the true velocity of A.

For the instant represented, link CB is rotating counterclockwise at a constant rate and its pin A causes a clockwise rotation of the slotted member ODE. Determine the angular velocity and angular acceleration of ODE for this instant. Problem 5/176 A D B

Cars A and B are both traveling on the curved intersecting roads with equal constant speeds of 30 mi/hr. For the positions shown, obtain the vector expressions for the velocity and acceleration which A appears to have to an observer in B who rotates with the car. The x-y axes are attached to car B. Problem 5/177

The disk rotates about a fixed axis through point O with a clockwise angular velocity and a counterclockwise angular acceleration at the instant under consideration. The value of r is 200 mm. Pin A is fixed to the disk but slides freely within the slotted member BC. Determine the velocity and acceleration of A relative to slotted member BC and the angular velocity and angular acceleration of BC. Problem 5/178

All conditions of the previous problem remain the same, except now, rather than rotating about a fixed center, the disk rolls without slipping on the horizontal surface. If the disk has a clockwise angular velocity of and a counterclockwise angular acceleration of determine the velocity and acceleration of pin A relative to the slotted member BC and the angular velocity and angular acceleration of BC. The value of r is 200 mm. Neglect the distance from the center of pin A to the edge of the disk. Problem 5/179

Two satellites are in circular equatorial orbits of different altitudes. Satellite A is in a geosynchronous orbit (one with the same period as the earths rotation so that it hovers over the same spot on the equator). Satellite B has an orbit of radius Calculate the velocity which A appears to have to an observer fixed in B when the elevation angle is (a) 0 and (b) The x-y axes are attached to B, whose antenna always points toward the center of the earth ( -direction). Consult and Appendix D for the necessary orbital information. Problem 5/180

The figure shows the vanes of a centrifugal-pump impeller which turns with a constant clockwise speed of . The fluid particles are observed to have an absolute velocity whose component in the r-direction is at discharge from the vane. Furthermore, the magnitude of the velocity of the particles measured relative to the vane is increasing at the rate of just before they leave the vane. Determine the magnitude of the total acceleration of a fluid particle an instant before it leaves the impeller. The radius of curvature of the vane at its end is 8 in. Problem 5/181

The crank OA revolves clockwise with a constant angular velocity of 10 rad/s within a limited arc of its motion. For the position determine the angular velocity of the slotted link CB and the acceleration of A as measured relative to the slot in CB. Problem 5/182

The Geneva wheel of Prob. 5/56 is shown again here. Determine the angular acceleration of wheel C for the instant when Wheel A has a constant clockwise angular velocity of 2 rad/s. Problem 5/183 200 mm P A C B O1 O2

The space shuttle A is in an equatorial circular orbit of 240-km altitude and is moving from west to east. Determine the velocity and acceleration which it appears to have to an observer B fixed to and rotating with the earth at the equator as the shuttle passes overhead. Use for the radius of the earth. Also use Fig. 1/1 for the appropriate value of g and carry out your calculations to 4-figure accuracy. Problem 5/184

The frictional resistance to the rotation of a flywheel consists of a retardation due to air friction which varies as the square of the angular velocity and a constant frictional retardation in the bearing. As a result the angular acceleration of the flywheel while it is allowed to coast is given by , where K and k are constants. Determine an expression for the time required for the flywheel to come to rest from an initial angular velocity

The wheel slips as it rolls. If and if the velocity of A with respect to B is locate the instantaneous center C of zero velocity and find the velocity of point P. Problem 5/186

The bar of is repeated here. If the velocity and tangential acceleration of end A are as indicated in the figure, determine the angular acceleration of the bar. Problem 5/187

The flywheel is rotating with an angular velocity at time when a torque is applied to increase its angular velocity. If the torque is controlled so that the angle between the total acceleration of point A on the rim and the radial line to A remains constant, determine the angular velocity and the angular acceleration as functions of the time t. Problem 5/188

The equilateral triangular plate is guided by the two vertex rollers A and B, which are confined to move in the perpendicular slots. The control rod gives A a constant velocity to the left for an interval of its motion. Determine the value of for which the horizontal component of the velocity of C is zero. Problem 5/189

Roller B of the linkage has a velocity of to the right as the angle passes and bar AB also makes an angle of with the horizontal. Locate the instantaneous center of zero velocity for bar AB and determine its angular velocity Problem 5/190

The pin A in the bell crank AOD is guided by the flanges of the collar B, which slides with a constant velocity of along the fixed shaft for an interval of motion. For the position determine the acceleration of the plunger CE, whose upper end is positioned by the radial slot in the bell crank. Problem 5/191 6 6 90 9

The helicopter is flying in the horizontal x-direction with a velocity and the plane of rotation of the 26-ft-diameter rotor is tilted from the horizontal x-y plane. The rotor blades rotate with an angular velocity For the instant represented write the vector expressions for the absolute velocities of rotor tip A and rotor tip B. Problem 5/192

The wheel rolls without slipping, and its position is controlled by the motion of the slider B. If B has a constant velocity of to the left, determine the angular velocity of AB and the velocity of the center O of the wheel when Problem 5/193

If the center O of the wheel of Prob. 5/193 has a constant velocity of to the left, calculate the acceleration of the slider B for the position 0.

In the linkage shown OC has a constant clockwise angular velocity during an interval of motion, while the hydraulic cylinder gives pin A a constant velocity of to the right. For the position shown where OC is vertical and BC is horizontal, calculate the angular velocity of BC. Solve by drawing the necessary velocity polygon. Problem 5/195

To speed up the unrolling of a telephone cable the trailer with the reel of cable starts from rest and is given an initial acceleration of Simultaneously, the tow truck pulls the free end of the cable horizontally in the opposite direction with an initial acceleration of If both vehicles start from rest at the same instant, determine the magnitude of the total acceleration of point A on the forward end of the horizontal reel diameter (a) just as the motion starts and (b) one second after the start of the motion. Problem 5/196

The hydraulic cylinder C imparts a velocity v to pin B in the direction shown. The collar slips freely on rod OA. Determine the resulting angular velocity of rod OA in terms of v, the displacement s of pin B, and the fixed distance d, for the angle Problem 5/197

The figure illustrates a commonly used quickreturn mechanism which produces a slow cutting stroke of the tool (attached to D) and a rapid return stroke. If the driving crank OA is turning at the constant rate determine the magnitude of the velocity of point B for the instant when Problem 5/198

The hydraulic cylinder moves pin A to the right with a constant velocity v. Use the fact that the distance from A to B is invariant, where B is the point on AC momentarily in contact with the gear, and write expressions for the angular velocity of the gear and the angular velocity of the rack AC. Problem 5/199

For the position shown where point A on the sliding collar has a constant velocity with corresponding lengthening of the hydraulic cylinder AC. For this same position BD is horizontal and DE is vertical. Determine the angular acceleration of DE at this instant. Problem 5/200

The tilting device maintains a sloshing water bath for washing vegetable produce. If the crank OA oscillates about the vertical and has a clockwise angular velocity of when OA is vertical, determine the angular velocity of the basket in the position shown where Problem 5/201

Determine the angular acceleration of the basket of the vegetable washer of for the position where OA is vertical. In this position OA has an angular velocity of and no angular acceleration

A radar station B situated at the equator observes a satellite A in a circular equatorial orbit of altitude and moving from west to east. For the instant when the satellite is above the horizon, determine the difference between the velocity of the satellite relative to the radar station, as measured from a nonrotating frame of reference, and the velocity as measured relative to the reference frame of the radar system. Problem 5/203 3

The crank OB revolves clockwise at the constant rate of For the instant when determine the angular acceleration of the rod BD, which slides through the pivoted collar at C. Problem 5/204

The disk rotates about a fixed axis with a constant angular velocity Pin A is fixed to the disk. Determine and plot the magnitudes of the velocity and acceleration of pin A relative to the slotted member BC as functions of the disk angle over the range State the maximum and minimum values and also the values of at which they occur. The value of r is 200 mm. Problem 5/205

Link OA is given a constant counterclockwise angular velocity Determine the angular velocity of link AB as a function of Compute and plot the ratio for the range Indicate the value of for which the angular velocity of AB is half that of OA.

The crank OA of the four-bar linkage is driven at a constant counterclockwise angular velocity Determine and plot as functions of the crank angle the angular velocities of bars AB and BC over the range State the maximum absolute value of each angular velocity and the value of at which it occurs. Problem 5/207

If all conditions in the previous problem remain the same, determine and plot as functions of the crank angle the angular accelerations of bars AB and BC over the range State the maximum absolute value of each angular acceleration and the value of at which it occurs.

All conditions of Prob. 5/207 remain the same, except the counterclockwise angular velocity of crank OA is when and the constant counterclockwise angular acceleration of the crank is Determine and plot as functions of the crank angle the angular velocities of bars AB and BC over the range State the maximum absolute value of each angular velocity and the value of at which it occurs.

For the Geneva wheel of Prob. 5/56, shown again here, write the expression for the angular velocity of the slotted wheel C during engagement of pin P and plot for the range The driving wheel A has a constant angular velocity Problem 5/210

The double crank is pivoted at O and permits complete rotation without interference with the pivoted rod CB as it slides through the collar A. If the crank has a constant angular velocity determine and plot the ratio as a function of between and By inspection determine the angle for which Problem 5/211

For the slider-crank configuration shown, derive the expression for the velocity of the piston (taken positive to the right) as a function of Substitute the numerical data of Sample Problem 5/15 and calculate as a function of for Plot versus and find its maximum magnitude and the corresponding value of . (By symmetry anticipate the results for Problem 5/212

For the slider-crank of derive the expression for the acceleration of the piston (taken positive to the right) as a function of for constant. Substitute the numerical data of Sample Problem and calculate as a function of for Plot versus and find the value of for which (By symmetry anticipate the results for 180   360).