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engineering
engineering mechanics dynamics

Engineering Mechanics Dynamics 8th Edition James L. Meriam, L. G. Kraige, J. N. Bolton - Solutions

In a study of head injury against the instrument panel of a car during sudden or crash stops where lap belts without shoulder straps or airbags are used, the segmented human model shown in the figure is analyzed. The hip joint O is assumed to remain fixed relative to the car, and the torso above
The 25-mm-thick uniform steel plate is being pulled very slowly into position by the cable connected to a winch. If the cable breaks at the instant represented, what are the force under each roller and the acceleration aG of the center of mass? Neglect friction and the mass of the rollers, and
The connecting rod AB of a certain internal-combustion engine weighs 1.2 lb with mass center at G and has a radius of gyration about G of 1.12 in. The piston and piston pin A together weigh 1.80 lb. The engine is running at a constant speed of 3000 rev/min, so that the angular velocity of the crank
The truck carries a 1500-mm-diameter spool of cable with a mass of 0.75 kg per meter of length. There are 150 turns on the full spool. The empty spool has a mass of 140 kg with radius of gyration of 530 mm. The truck alone has a mass of 2030 kg with mass center at G. If the truck starts from rest
The four-bar mechanism lies in a vertical plane and is controlled by crank OA which rotates counterclockwise at a steady rate of 60 rev /min. Determine the torque M which must be applied to the crank at O when the crank angle θ = 45°. The uniform coupler AB has a mass of 7 kg, and the masses of
The Ferris wheel at an amusement park has an even number n of gondolas, each freely pivoted at its point of support on the wheel periphery. Each gondola has a loaded mass m, a radius of gyration k about its point of support A, and a mass center a distance h from A. The wheel structure has a moment
The slender rod of mass m and length l has a particle (negligible radius, mass 2m) attached to its end. If the body is released from rest when in the position shown, determine its angular velocity as it passes the vertical position. 2m т 31 4 17
The log is suspended by the two parallel 5-m cables and used as a battering ram. At what angle θ should the log be released from rest in order to strike the object to be smashed with a velocity of 4 m/s? 5 m 5 m L-- v = 4 m/s
The assembly is constructed of homogeneous slender rod which has a mass ρ per unit length, and it rotates freely about a horizontal axis through the pivot at O. If the assembly is nudged from the starting position shown, determine its angular velocity after it has rotated (a) 45°, (b)
The velocity of the 8-kg cylinder is 0.3 m/s at a certain instant. What is its speed v after dropping an additional 1.5 m? The mass of the grooved drum is 12 kg, its centroidal radius of gyration is k̅ = 210 mm, and the radius of its groove is ri = 200 mm. The frictional moment at O is a constant
The 32.2-lb wheel is released from rest and rolls on its hubs without slipping. Calculate the speed v of the center O of the wheel after it has moved a distance x = 10 ft down the incline. The radius of gyration of the wheel about O is 5 in. 2" 8" 1. 10
The uniform semicircular bar of radius r = 75 mm and mass m = 3 kg rotates freely about a horizontal axis through the pivot O. The bar is initially held in position 1 against the action of the torsional spring and then suddenly released. Determine the spring stiffness kT which will give the bar a
The homogeneous rectangular crate weighs 250 lb and is supported in the horizontal position by the cable at A and the corner hinge at O. If the cable at A is suddenly released, calculate the angular velocity w of the crate just before it strikes the 30° incline. Does the weight of the crate
The 24-lb disk is rigidly attached to the 7-lb bar OA, which is pivoted freely about a horizontal axis through point O. If the system is released from rest in the position shown, determine the angular velocity of the bar and the magnitude of the pin reaction at O after the bar has rotated 90°. A
The two wheels of Prob. 6 /78, shown again here, represent two extreme conditions of distribution of mass. For case A all of the mass m is assumed to be concentrated in the center of the hoop in the axial bar of negligible diameter. For case B all of the mass m is assumed to be concentrated in the
The 15-kg slender bar OA is released from rest in the vertical position and compresses the spring of stiffness k = 20 kN/m as the horizontal position is passed. Determine the proper setting of the spring, by specifying the distance h, which will result in the bar having an angular velocity w = 4
The light circular hoop of radius r contains a quarter-circular sector of mass m and is initially at rest on the horizontal surface. A couple M is applied to the hoop, which rolls without slipping along the horizontal surface. Determine the velocity v of the hoop after it has rolled through
A steady 5-lb force is applied normal to the handle of the hand-operated grinder. The gear inside the housing with its shaft and attached handle together weigh 3.94 lb and have a radius of gyration about their axis of 2.85 in. The grinding wheel with its attached shaft and pinion (inside housing)
The figure shows an impact tester used in studying material response to shock loads. The 60-lb pendulum is released from rest and swings through the vertical with negligible resistance. At the bottommost point of the ensuing motion, the pendulum strikes a notched material specimen A. After impact
The uniform rectangular plate is released from rest in the position shown. Determine the maximum angular velocity w during the ensuing motion. Friction at the pivot is negligible. 26
The 50-kg flywheel has a radius of gyration k̅ = 0.4 m about its shaft axis and is subjected to the torque M = 2(1 − e−0.1θ) N∙m, where θ is in radians. If the flywheel is at rest when θ = 0, determine its angular velocity after 5 revolutions. M-
The figure shows the cross section AB of a garage door which is a rectangular 2.5-m by 5-m panel of uniform thickness with a mass of 200 kg. The door is supported by the struts of negligible mass and hinged at O. Two spring-and-cable assemblies, one on each side of the door, control the movement.
The uniform 40-lb bar with attached 12-lb wheels is released from rest when θ = 60°. If the wheels roll without slipping on the horizontal and vertical surfaces, determine the angular velocity of the bar when θ = 45°. Each wheel has a centroidal radius of gyration of 4.5 inches. B 6" 4' A
A 1200-kg flywheel with a radius of gyration of 400 mm has its speed reduced from 5000 to 3000 rev/min during a 2-min interval. Calculate the average power supplied by the flywheel. Express your answer both in kilowatts and in horsepower.
The wheel consists of a 4-kg rim of 250-mm radius with hub and spokes of negligible mass. The wheel is mounted on the 3-kg yoke OA with mass center at G and with a radius of gyration about O of 350 mm. If the assembly is released from rest in the horizontal position shown and if the wheel rolls on
The uniform slender bar ABC weighs 6 lb and is initially at rest with end A bearing against the stop in the horizontal guide. When a constant couple M = 72 lb-in. is applied to end C, the bar rotates causing end A to strike the side of the vertical guide with a velocity of 10 ft /sec. Calculate the
The torsional spring at A has a stiffness kT = 10 N∙m/rad and is undeformed when the uniform 10-kg bars OA and AB are in the vertical position and overlap. If the system is released from rest with θ = 60°, determine the angular velocity of wheel B when θ = 30°. The 6-kg wheel at B has a
The system is released from rest when the angle θ = 90°. Determine the angular velocity of the uniform slender bar when θ equals 60°. Use the values m1 = 1 kg, m2 = 1.25 kg, and b = 0.4 m. -2b- C A m1 m2 26 B
The homogeneous torus and cylindrical ring are released from rest and roll without slipping down the incline. Determine an expression for the velocity difference vdiff which develops between the two objects during the ensuing motion as a function of the distance x they have traveled down the
The uniform 12-lb disk pivots freely about a horizontal axis through O. A 4-lb slender bar is fastened to the disk as shown. If the system is nudged from rest while in the position shown, determine its angular velocity w after it has rotated 180°. 8". 30° 30°
Under active development is the storage of energy in high-speed rotating disks where friction is effectively eliminated by encasing the rotor in an evacuated enclosure and by using magnetic bearings. For a 10-kg rotor with a radius of gyration of 90 mm rotating initially at 80 000 rev/min,
For the pivoted slender rod of length l, determine the distance x for which the angular velocity will be a maximum as the bar passes the vertical position after being released in the horizontal position shown. State the corresponding angular velocity. -1-
The wheel has mass m and a centroidal radius of gyration k and rolls without slipping up the incline under the action of a force P. The force is applied to the end of a cord which is wrapped securely around the inner hub of the wheel as shown. Determine the speed vO of the wheel center O after the
Motion of the bar is controlled by the constrained paths of A and B. If the angular velocity of the bar is 2 rad/s counterclockwise as the position θ = 45° is passed, determine the speeds of points A and P. РО B 2 rad/s A 500 mm 500 mm
The shaft of the wheel unit rolls without slipping on the fixed horizontal surface, and point O has a velocity of 3 ft /sec to the right. By the method of this article, determine the velocities of points A, B, C, and D. A y | Lー-x 10" 2" vo = 3 ft/sec C D B
The center D of the car follows the centerline of the 100-ft skidpad. The speed of point D is v = 45 ft/sec. Determine the angular velocity of the car and the speeds of points A and B of the car. A 16' 100 ft 6'
The attached wheels roll without slipping on the plates A and B, which are moving in opposite directions as shown. If vA = 60 mm/s to the right and vB = 200 mm/s to the left, determine the speeds of the center O and the point P for the position shown. A 40 90 mm mm B
The mechanism of Prob. 5/78 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 w0 and r to be known. A 2r B 45° og 30°
At the instant under consideration, the rod of the hydraulic cylinder is extending at the rate vA = 2 m/s. Determine the corresponding angular velocity wOB of link OB. A VA = 2 m/s 60° 180 mm 120 mm B 15°
End A of the slender pole is given a velocity vA to the right along the horizontal surface. Show that the magnitude of the velocity of end B equals vA when the midpoint M of the pole comes in contact with the semicircular obstruction. В M A VA
The flexible band F is attached at E to the rotating sector and leads over the guide pulley G. Determine the angular velocities of links AB and BD for the position shown if the band has a speed of 2 m/s. 2 m/s OD 0.18 m 15°. 0.3 m 0.4 m G B 0.36 m A F E
The rear driving wheel of a car has a diameter of 26 in. and has an angular speed N of 200 rev/min 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 vs of the
Link OA has a counterclockwise angular velocity θ˙ = 4 rad/sec during an interval of its motion. Determine the angular velocity of link AB and of sector BD for θ = 45° at which instant AB is horizontal and BD is vertical. D 6" B 16" 8"
Vertical oscillation of the spring-loaded plunger F is controlled by a periodic change in pressure in the vertical hydraulic cylinder E. For the position θ = 60°, determine the angular velocity of AD and the velocity of the roller A in its horizontal guide if the plunger F has a downward velocity
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 v = 0.2 m/s. Determine the velocity of A when θ becomes 90°. E D ACWww 120 mm 60 mm Ww 06
A device which tests the resistance to wear of two materials A and B is shown. If the link EO has a velocity of 4 ft/sec to the right when θ = 45°, determine the rubbing velocity vA. D -5"→ 12" E 6" A
In the design of the mechanism shown, collar A is to slide along the fixed shaft as angle θ increases. When θ = 30°, the control link at D is to have a downward component of velocity of 0.60 m/s. Determine the corresponding velocity of collar A by the method of this article. A 700 mm -800 mm B
Determine the angular velocity w of the ram head AE of the rock crusher in the position for which θ = 60°. The crank OB has an angular speed of 60 rev/min. 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 OB =
For the instant represented, corner C of the rectangular plate has an acceleration of 5 m/s2 in the negative y-direction, and the plate has a clockwise angular velocity of 4 rad/s which is decreasing by 12 rad /s each second. Determine the magnitude of the acceleration of A at this instant. Solve
The two rotor blades of 800-mm radius rotate counterclockwise with a constant angular velocity w = θ˙ = 2 rad/s about the shaft at O mounted in the sliding block. The acceleration of the block is aO = 3 m/s2. Determine the magnitude of the acceleration of the tip A of the blade when (a) θ =
A container for waste materials is dumped by the hydraulically-activated linkage shown. If the piston rod starts from rest in the position indicated and has an acceleration of 1.5 ft/sec2 in the direction shown, compute the initial angular acceleration of the container. 6'- B- A 45° Chute
Determine the angular velocity and angular acceleration of the slender bar AB just after roller B passes point C and enters the circular portion of the support surface. R A В C L-
The wheel of radius R rolls without slipping, and its center O has an acceleration aO. A point P on the wheel is a distance r from O. For given values of aO, R, and r, determine the angle θ and the velocity vO of the wheel for which P has no acceleration in this position. R vo ao – lo P
The 9-m steel beam is being hoisted from its horizontal position by the two cables attached at A and B. If the initial angular accelerations of the hoisting drums are a1 = 0.5 rad/s2 and a2 = 0.2 rad/s2 in the directions shown, determine the corresponding angular acceleration a of the beam, the
For a short interval, collars A and B are sliding along the fixed vertical shaft with velocities vA = 2 m/s and vB = 3 m/s in the directions shown. Determine the magnitude of the velocity of point C for the position θ = 60°. A VA 400 mm 400 mm UB B
The right-angle link has a counterclockwise angular velocity of 3 rad /s at the instant represented, and point B has a velocity vB = 2i − 0.3j m/s. Determine the velocity of A using vector notation. Sketch the vector polygon which corresponds to the terms in the relative-velocity equation and
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 v = 3 ft/sec, determine the magnitudes of the velocity of point O and the velocity of point B on the larger
The rider of the bicycle shown pumps steadily to maintain a constant speed of 16 km/h against a slight head wind. Calculate the maximum and minimum magnitudes of the absolute velocity of the pedal A. 660 mm /70 mm 170 160 mm mm
Determine the angular velocity of bar AB just after roller B has begun moving up the 15° incline. At the instant under consideration, the velocity of roller A is vA. VA L 15° A
The spoked wheel of radius r is made to roll up the incline by the cord wrapped securely around a shallow groove on its outer rim. For a given cord speed v at point P, determine the velocities of points A and B. No slipping occurs. y B A
At the instant represented, the velocity of point A of the 1.2-m bar is 3 m/s to the right. Determine the speed vB of point B and the angular velocity w of the bar. The diameter of the small end wheels may be neglected. 0.5 m 60° B 1.2 m А VA
Determine the angular velocity of link BC for the instant indicated. In case (a), the center O of the disk is a fixed pivot, while in case (b), the disk rolls without slipping on the horizontal surface. In both cases, the disk has clockwise angular velocity w. Neglect the small distance of pin A
The elements of a switching device are shown. If the vertical control rod has a downward velocity v = 2 ft/sec when the device is in the position shown, determine the corresponding speed of point A. Roller C is in continuous contact with the inclined surface. A O 3" 30° 3" В C 15°
Determine the angular velocity wAB of link AB and the velocity vB of collar B for the instant represented. Assume the quantities w0 and r to be known. A 2r 45° B
Determine the angular velocity wAB of link AB and the velocity vB of collar B for the instant represented. Assume the quantities w0 and r to be known. A 2r B 45° 30°
The rotation of the gear is controlled by the horizontal motion of end A of the rack AB. If the piston rod has a constant velocity x˙ = 300 mm/s during a short interval of motion, determine the angular velocity w0 of the gear and the angular velocity wAB of AB at the instant when x = 800 mm. B A
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 wP = 2 rad/s. Determine the corresponding angular velocities of the two
The elements of a simplified clam-shell bucket for a dredge are shown. The cable which opens and closes the bucket passes through the block at O. With O as a fixed point, determine the angular velocity w of the bucket jaws when θ = 45° as they are closing. The upward velocity of the control cable
In the design of a produce-processing plant, roller trays of produce are to be oscillated under water spray by the action of the connecting link AB and crank OB. For the instant when θ = 15°, the angular velocity of AB is 0.086 rad /s clockwise. Find the corresponding angular velocity θ˙ of the
The vertical rod has a downward velocity v = 0.8 m/s when link AB is in the 30° position shown. Determine the corresponding angular velocity of AB and the speed of roller B if R = 0.4 m. 1.25R R 30° B
Pin P on the end of the horizontal rod slides freely in the slotted gear. The gear engages the moving rack A and the fixed rack B (teeth not shown) so it rolls without slipping. If A has a velocity of 0.4 ft/sec to the left for the instant shown, determine the velocity vP of the rod for this
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 w0 = 10 rad/s, determine the angular velocities of links AB and BC. B, 15° 240 mm A 80 mm 200 60° mm
The mechanism is part of a latching device where rotation of link AOB is controlled by the rotation of slotted link D about C. If member D has a clockwise angular velocity of 1.5 rad /s when the slot is parallel to OC, determine the corresponding angular velocity of AOB. Solve graphically or
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 wOB = 0.5 rad/sec if tan θ = 4/3 at this instant. y A B WOB 6" 10" -6"
The wheel rolls without slipping. For the instant portrayed, when O is directly under point C, link OA has a velocity v = 1.5 m/s to the right and θ = 30°. Determine the angular velocity w of the slotted link. Р 200 P mm 100 mm A
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 speeds of points A and B. 4 rad/s 2 m/s AC 0.3 m G 0.3 m
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. 0.3 m 4 rad/s G 20° 0.3 m 2 m/s A
For the instant represented, corner A of the rectangular plate has a velocity vA = 2.8 m/s and the plate has a clockwise angular velocity w = 12 rad/s. Determine the magnitude of the corresponding velocity of point B. VA 30° A 120 mm 160 mm
Roller B of the quarter-circular link has a velocity vB = 3 ft /sec directed down the 15° incline. The link has a counterclockwise angular velocity w = 2 rad /sec. By the method of this article, determine the velocity of roller A. A 6" UB 15°
The bar AB has a counterclockwise angular velocity of 6 rad/sec. Construct the velocity vectors for points A and G of the bar and specify their magnitudes if the instantaneous center of zero velocity for the bar is (a) At C1, (b) At C2, (c) At C3. - 4". C2 C1 G 3" B. 1" 1" 3" C3
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
At a certain instant vertex B of the right-triangular plate has a velocity of 200 mm/s in the direction shown. If the instantaneous center of zero velocity for the plate is 40 mm from point B and if the angular velocity of the plate is clockwise, determine the speed of point D. 40 mm 30 mm UB
End A of the 2-ft link has a velocity of 5 ft/sec in the direction shown. At the same instant, end B has a velocity whose magnitude is 7 ft/sec as indicated. Find the angular velocity ω of the link in two ways. VA = 5 ft/sec 40° AQ 2' В B VB = 7 ft/sec
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, (c) Equal to 8 ft/sec to the right.
The uniform rectangular plate moves on the horizontal surface. Its mass center has a velocity vG = 3 m/s directed parallel to the x-axis and the plate has a counterclockwise (as seen from above) angular velocity ω = 4 rad/s. Determine the velocities of points A and B. -0.3 m UG G В -0.5 m
The right-angle link AB has a clockwise angular velocity ω = 2 rad/sec when in the position shown. Determine the velocity of B with respect to A for this instant. A y 30° 12" Lーーズ 5" B の
The punch is operated by a simple harmonic oscillation of the pivoted sector given by θ = θ0 sin 2πt where the amplitude is θ0 = π/12 rad (15°) and the time for one complete oscillation is 1 second. Determine the acceleration of the punch when (a) θ = 0 and (b) θ = π/12. 140 mm 100 mm
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, θ = 45°. For a
One of the most common mechanisms is the slider-crank. Express the angular velocity ωAB and angular acceleration αAB of the connecting rod AB in terms of the crank angle θ for a given constant crank speed ω0. Take ωAB and αAB to be positive counterclockwise. B А
Angular oscillation of the slotted link is achieved by the crank OA, which rotates clockwise at the steady speed N = 120 rev/min. Determine an expression for the angular velocity β˙ of the slotted link in terms of θ. А 2.5" B 9"
A variable-speed belt drive consists of the two pulleys, each of which is constructed of two cones which turn as a unit but are capable of being drawn together or separated so as to change the effective radius of the pulley. If the angular velocity ω1 of pulley 1 is constant, determine the
The rotation of link AO is controlled by the piston rod of hydraulic cylinder BC, which is elongating at the constant rate s˙ = k for an interval of motion. Write the vector expression for the acceleration of end A for a given value of θ using unit vectors en and et with n − t coordinates. A n
Show that the expressions v = rω and at = rα hold for the motion of the center O of the wheel which rolls on the concave or convex circular arc, where ω and α are the absolute angular velocity and acceleration, respectively, of the wheel. R -t R
It is desired to design a system for controlling the rate of extension x˙ of the fire-truck ladder during elevation of the ladder so that the bucket B will have vertical motion only. Determine x˙ in terms of the elongation rate c˙ of the hydraulic cylinder for given values of θ and x. Horiz. A
Derive an expression for the upward velocity v of the car hoist in terms of θ. The piston rod of the hydraulic cylinder is extending at the rate s˙. b. 26 b. L
Link OA is given a clockwise angular velocity ω = 2 rad/sec as indicated. Determine the velocity v of point C for the position θ = 30° if b = 8 in. А b 26 B
Motion of the wheel as it rolls up the fixed rack on its geared hub is controlled through the peripheral cable by the driving wheel D, which turns counterclockwise at the constant rate ω0 = 4 rad/s for a short interval of motion. By examining the geometry of a small (differential) rotation of line
A roadway speed bump is being installed on a level road to remind motorists of the existing speed limit. If the driver of the car experiences at G a vertical acceleration of as much as g, up or down, he is expected to realize that his speed is bordering on being excessive. For the speed bump with
The rod OB slides through the collar pivoted to the rotating link at A. If CA has an angular velocity ω = 3 rad/s for an interval of motion, calculate the angular velocity of OB when θ = 45°. CA = 200 mm B A 400 mm
Vertical motion of the work platform is controlled by the horizontal motion of pin A. If A has a velocity v0 to the left, determine the vertical velocity v of the platform for any value of θ. L/2 L/2 L/2 e L12 A

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