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engineering
engineering mechanics dynamics

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

The hydraulic cylinder imparts a constant upward velocity vA = 0.2 m/s to corner A of the rectangular container during an interval of its motion. For the instant when θ = 20°, determine the velocity and acceleration of roller B. Also, determine the corresponding angular velocity of edge CD. D 0.5
Boom OA is being elevated by the rope-and-pulley arrangement shown. If point B on the rope is given a constant velocity vB = 9 ft/sec, determine the angular velocity and angular acceleration ω of the boom for θ = 30°. y A 18 ft B 30 ft UB
The collar C moves to the left on the fixed guide with speed v. Determine the angular velocity ωOA as a function of v, the collar position s, and the height h. A, В h C U-
Link OA rotates with a counterclockwise angular velocity ω = 3 rad/s. Determine the angular velocity of bar BC when θ = 20°. A b. В 26-
Determine the acceleration of the shaft B for θ = 60° if the crank OA has an angular acceleration θ¨ = 8 rad/s2 and an angular velocity θ˙ = 4 rad/s at this position. The spring maintains contact between the roller and the surface of the plunger. В 20 mm AO 80 mm-
Link OA rotates with a clockwise angular velocity ω = 7 rad/s. Determine the velocity of point B for the position θ = 30°. Use the values b = 80 mm, d = 100 mm, and h = 30 mm. A d h B
The wheel of radius r rolls without slipping, and its center O has a constant velocity vO 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
A rocket stage designed for deep-space missions consists of 200 kg of fuel and 300 kg of structure and payload combined. In terms of burnout velocity, what would be the advantage of reducing the structural/payload mass by 1 percent (3 kg) and using that mass for additional fuel? Express your answer
The upper end of the open-link chain of length L and mass ρ per unit length is lowered at a constant speed v by the force P. Determine the reading R of the platform scale in terms of x. P A L
The magnetometer boom for a spacecraft consists of a large number of triangular-shaped units which spring into their deployed configuration upon release from the canister in which they were folded and packed prior to release. Write an expression for the force F which the base of the canister must
A model rocket weighs 1.5 lb just before its vertical launch. Its experimental solid-fuel motor carries 0.1 lb of fuel, has an escape velocity of 3000 ft/sec, and burns the fuel for 0.9 sec. Determine the acceleration of the rocket at launch and its burnout velocity. Neglect aerodynamic drag and
A tank truck for washing down streets has a total weight of 20,000 lb when its tank is full. With the spray turned on, 80 lb of water per second issue from the nozzle with a velocity of 60 ft/sec relative to the truck at the 30° angle shown. If the truck is to accelerate at the rate of 2 ft/sec2
The mass m of a raindrop increases as it picks up moisture during its vertical descent through still air. If the air resistance to motion of the drop is R and its downward velocity is v, write the equation of motion for the drop and show that the relation ∑F = d(mv)/dt is obeyed as a special case
A small rocket of initial mass m0 is fired vertically upward near the surface of the earth (g constant). If air resistance is neglected, determine the manner in which the mass m of the rocket must vary as a function of the time t after launching in order that the rocket may have a constant vertical
The space shuttle, together with its central fuel tank and two booster rockets, has a total mass of 2.04(106) kg at liftoff. Each of the two booster rockets produces a thrust of 11.80(106) N, and each of the three main engines of the shuttle produces a thrust of 2.00(106) N. The specific impulse
One of the functions of the space shuttle is to release communications satellites at low altitude. A booster rocket is fi red at B, placing the satellite in an elliptical transfer orbit, the apogee of which is at the altitude necessary for a geosynchronous orbit. (A geosynchronous orbit is an
The angular velocity of a gear is controlled according to ω = 12 − 3t2 where ω, in radians per second, is positive in the clockwise sense and where t is the time in seconds. Find the net angular displacement ∆θ from the time t = 0 to t = 3 s. Also find the total number of revolutions N
Magnetic tape is fed over and around the light pulleys mounted in a computer frame. If the speed v of the tape is constant and if the ratio of the magnitudes of the acceleration of points A and B is 2/3, determine the radius r of the larger pulley. B 3" А
When switched on, the grinding machine accelerates from rest to its operating speed of 3450 rev/min in 6 seconds. When switched off, it coasts to rest in 32 seconds. Determine the number of revolutions turned during both the startup and shutdown periods. Assume uniform angular acceleration in both
The small cart is released from rest in position 1 and requires 0.638 seconds to reach position 2 at the bottom of the path, where its center G has a velocity of 14.20 ft/sec. Determine the angular velocity ω of line AB in position 2 and the average angular velocity ωav of AB during the interval.
The flywheel has a diameter of 600 mm and rotates with increasing speed about its z-axis shaft. When point P on the rim crosses the y-axis with θ = 90°, it has an acceleration given by a = −1.8i − 4.8j m /s2. For this instant, determine the angular velocity ω and the angular acceleration α
The drive mechanism imparts to the semicircular plate simple harmonic motion of the form θ = θ0 sin ω0t, where θ0 is the amplitude of the oscillation and ω0 is its circular frequency. Determine the amplitudes of the angular velocity and angular acceleration and state where in the motion cycle
The cylinder rotates about the fixed z-axis in the direction indicated. If the speed of point A is vA = 2 ft/sec and the magnitude of its acceleration is aA = 12 ft/sec2, determine the angular velocity and angular acceleration of the cylinder. Is knowledge of the angle θ necessary? 6 A 一
The angular acceleration of a body which is rotating about a fixed axis is given by α = −kw2, where the constant k = 0.1 (no units). Determine the angular displacement and time elapsed when the angular velocity has been reduced to one-third its initial value ω0 = 12 rad /s.
The device shown rotates about the fixed z-axis with angular velocity ω = 20 rad /s and angular acceleration α = 40 rad/s2 in the directions indicated. Determine the instantaneous velocity and acceleration of point B. B 60°- 60° A y
The circular disk rotates with a constant angular velocity ω = 40 rad /sec about its axis, which is inclined in the y-z plane at the angle θ = tan−1 3/4. Determine the vector expressions for the velocity and acceleration of point P, whose position vector at the instant shown is r = 15i +
The T-shaped body rotates about a horizontal axis through point O. At the instant represented, its angular velocity is ω = 5 rad/sec and its angular acceleration is α = 10 rad/sec2 in the directions indicated. Determine the velocity and acceleration of (a) point A and (b) point B. Express your
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 t = 0 and uniformly accelerates with α = 2 rad/s2. If the adhesive fails at
The two attached pulleys are driven by the belt with increasing speed. When the belt reaches a speed v = 2 ft/sec, the total acceleration of point P is 26 ft/sec2. For this instant determine the angular acceleration α of the pulleys and the acceleration of point B on the belt. 8" B A 8"
The bent flat bar rotates about a fixed axis through point O with the instantaneous angular properties indicated in the figure. Determine the velocity and acceleration of point A. o = 2 rad/s a = 5 rad/s? y 0.5 m L--x 15°- -x- 120° 0.3 m PA
At time t = 0, the arm is rotating about the fixed z-axis with an angular velocity ω = 200 rad/s in the direction shown. At that time, a constant angular deceleration begins and the arm comes to a stop in 10 seconds. At what time t does the acceleration of point P make a 15° angle with the arm
A variable torque is applied to a rotating wheel at time t = 0 and causes the clockwise angular acceleration to increase linearly with the clockwise angular displacement θ of the wheel during the next 30 revolutions. When the wheel has turned the additional 30 revolutions, its angular velocity is
Develop general expressions for the instantaneous velocity and acceleration of point A of the square plate, which rotates about a fixed axis through point O. Take all variables to be positive. Then evaluate your expressions for θ = 30°, b = 0.2 m, ω = 1.4 rad/s, and α = 2.5 rad/s2. b. y 1. L--x
The motor A accelerates uniformly from zero to 3600 rev/min in 8 seconds after it is turned on at time t = 0. It drives a fan (not shown) which is attached to drum B. The effective pulley radii are shown in the figure. Determine (a) The number of revolutions turned by drum B during the
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
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 t = 2 s to give gear A a counterclockwise angular acceleration α which varies with time for a duration of 4 seconds as shown.
A V-belt speed-reduction drive is shown where pulley A drives the two integral pulleys B which in turn drive pulley C. If A starts from rest at time t = 0 and is given a constant angular acceleration a1, derive expressions for the angular velocity of C and the magnitude of the acceleration of a
Slider A moves in the horizontal slot with a constant speed v for a short interval of motion. Determine the angular velocity ω of bar AB in terms of the displacement xA. 40° A XA-
The concrete pier P is being lowered by the pulley and cable arrangement shown. If points A and B have velocities of 0.4 m/s and 0.2 m/s, respectively, compute the velocity of P, the velocity of point C for the instant represented, and the angular velocity of the pulley. VA UB A 200 mm 100 mm P
At the instant under consideration, the hydraulic cylinder AB has a length L = 0.75 m, and this length is momentarily increasing at a constant rate of 0.2 m /s. If vA = 0.6 m /s and θ = 35°, determine the velocity of slider B. B L VA A
The hydraulic cylinder D is causing the distance OA to increase at the rate of 2 in./sec. Calculate the velocity of the pin at C in its horizontal guide for the instant when θ = 50o. AO 6". 6". B D 6" C
The Scotch-yoke mechanism converts rotational motion of the disk to oscillatory translation of the shaft. For given values of θ, ω, α, r, and d, determine the velocity and acceleration of point P of the shaft. A r. aY -d-
The Scotch-yoke mechanism of Prob. 5/34 is modified as shown in the figure. For given values of ω, α, r, θ, d, and β, determine the velocity and acceleration of point P of the shaft. at P d- B
At a bulk loading station, gravel leaves the hopper at the rate of 220 lb/sec with a velocity of 10 ft/sec in the direction shown and is deposited on the moving flatbed truck. The tractive force between the driving wheels and the road is 380 lb, which overcomes the 200 lb of frictional road
A railroad coal car weighs 54,600 lb empty and carries a total load of 180,000 lb of coal. The bins are equipped with bottom doors which permit discharging coal through an opening between the rails. If the car dumps coal at the rate of 20,000 lb/sec in a downward direction relative to the car, and
A coil of heavy flexible cable with a total length of 100 m and a mass of 1.2 kg/m is to be laid along a straight horizontal line. The end is secured to a post at A, and the cable peels off the coil and emerges through the horizontal opening in the cart as shown. The cart and drum together have a
By lowering a scoop as it skims the surface of a body of water, the aircraft (nicknamed the “Super Scooper”) is able to ingest 4.5 m3 of fresh water during a 12-second run. The plane then flies to a fire area and makes a massive water drop with the ability to repeat the procedure as many times
A small rocket-propelled vehicle weighs 125 lb, including 20 lb of fuel. Fuel is burned at the constant rate of 2 lb/sec with an exhaust velocity relative to the nozzle of 400 ft/sec. Upon ignition the vehicle is released from rest on the 10° incline. Calculate the maximum velocity v reached by
The end of a pile of loose-link chain of mass ρ per unit length is being pulled horizontally along the surface by a constant force P. If the coefficient of kinetic friction between the chain and the surface is μk, determine the acceleration a of the chain in terms of x and x˙.
A coal car with an empty mass of 25 Mg is moving freely with a speed of 1.2 m/s under a hopper which opens and releases coal into the moving car at the constant rate of 4 Mg per second. Determine the distance x moved by the car during the time that 32 Mg of coal are deposited in the car. Neglect
Sand is released from the hopper H with negligible velocity and then falls a distance h to the conveyor belt. The mass flow rate from the hopper is m′. Develop an expression for the steady-state belt speed v for the case h = 0. Assume that the sand quickly acquires the belt velocity with no
The open-link chain of length L and mass ρ per unit length is released from rest in the position shown, where the bottom link is almost touching the platform and the horizontal section is supported on a smooth surface. Friction at the corner guide is negligible. Determine (a) The velocity v1
In the figure is shown a system used to arrest the motion of an airplane landing on a field of restricted length. The plane of mass m rolling freely with a velocity v0 engages a hook which pulls the ends of two heavy chains, each of length L and mass ρ per unit length, in the manner shown. A
The free end of the flexible and inextensible rope of mass ρ per unit length and total length L is given a constant upward velocity v. Write expressions for P, the force R supporting the fixed end, and the tension T1 in the rope at the loop in terms of x. (For the loop of negligible size, the
The system of three particles has the indicated particle masses, velocities, and external forces. Determine r̅, r̅̇, r̅̈ , T, HO, and H˙ O for this three-dimensional system. 4m 1.5d F 3v d 2d m 2v -y 2m
Each of the identical steel balls weighs 4 lb and is fastened to the other two by connecting bars of negligible weight and unequal length. In the absence of friction at the supporting horizontal surface, determine the initial acceleration a̅ of the mass center of the assembly when it is subjected
A 2-oz bullet is fired horizontally with a velocity v = 1000 ft/sec into the slender bar of a 3-lb pendulum initially at rest. If the bullet embeds itself in the bar, compute the resulting angular velocity of the pendulum immediately after the impact. Treat the sphere as a particle and neglect the
A large rocket ready for vertical launch has a total mass of 2.7(103) Mg. At launch, fuel is burned at the rate of 13 Mg/s with an exhaust velocity of 2400 m/s. Determine the initial acceleration a of the rocket. Assume atmospheric pressure across the exit plane of the exhaust nozzle.
In an operational design test of the equipment of the fire truck, the water cannon is delivering fresh water through its 2-in.-diameter nozzle at the rate of 1400 gal/min at the 20° angle. Calculate the total friction force F exerted by the pavement on the tires of the truck, which remains in a
The rocket shown is designed to test the operation of a new guidance system. When it has reached a certain altitude beyond the effective influence of the earth’s atmosphere, its mass has decreased to 2.80 Mg, and its trajectory is 30° from the vertical. Rocket fuel is being consumed at the rate
When only the air of a sand-blasting gun is turned on, the force of the air on a flat surface normal to the stream and close to the nozzle is 20 N. With the nozzle in the same position, the force increases to 30 N when sand is admitted to the stream. If sand is being consumed at the rate of 4.5
A two-stage rocket is fired vertically up and is above the atmosphere when the first stage burns out and the second stage separates and ignites. The second stage carries 1200 kg of fuel and has an empty mass of 200 kg. Upon ignition the second stage burns fuel at the rate of 5.2 kg/s and has a
Water of density ρ issues from the nozzle of area A and impinges upon the block of mass m, which is at rest on the rough horizontal surface. If the block is on the verge of tipping rightward, determine the required minimum coefficient of friction μ between the block and the surface and the exit
The block of Prob. 4 /102 is now fitted with two small rollers. Determine the normal force under each roller and the initial rightward acceleration of the block if the exit velocity of the water is (a) 75 percent of the tipping velocity (b) 50 percent of the tipping velocity. Refer
A jet of fresh water under pressure issues from the 3/4-in.-diameter fixed nozzle with a velocity v = 120 ft/sec and is diverted into the two equal streams. Neglect any energy loss in the streams and compute the force F required to hold the vane in place. 30° A F 30°
The flexible nonextensible rope of length πr/2 and mass ρ per unit length is attached at A to the fixed quarter-circular guide and allowed to fall from rest in the horizontal position. When the rope comes to rest in the dashed position, the system will have lost energy. Determine the loss ΔQ and
In the static test of a jet engine and exhaust nozzle assembly, air is sucked into the engine at the rate of 30 kg /s and fuel is burned at the rate of 1.6 kg/s. The flow area, static pressure, and axial-flow velocity for the three sections shown are as follows:Sec. A Sec. B Sec. CFlow area, m2
The upper end of the open-link chain of length L and mass ρ per unit length is released from rest with the lower end just touching the platform of the scale. Determine the expression for the force F read on the scale as a function of the distance x through which the upper end has fallen. (Comment:
The chain of length L and mass ρ per unit length is released from rest on the smooth horizontal surface with a negligibly small overhang x to initiate motion. Determine (a) The acceleration a as a function of x.(b) The tension T in the chain at the smooth corner as a function of x.(c) The
A vertical force P acting on the 2.5-lb cylindrical valve A, shown in section, serves to limit the flow of fresh water from the top of the vertical pipe B of 3-in. inside diameter. Water is fed through the bottom inlet of the pipe. Calculate the force P required to maintain the valve in the
The chain of mass ρ per unit length passes over the small freely turning pulley and is released from rest with only a small imbalance h to initiate motion. Determine the acceleration a and velocity v of the chain and the force R supported by the hook at A, all in terms of h as it varies from
The centrifugal pump handles 20 m3 of fresh water per minute with inlet and outlet velocities of 18 m/s. The impeller is turned clockwise through the shaft at O by a motor which delivers 40 kW at a pump speed of 900 rev/min. With the pump filled but not turning, the vertical reactions at C and D
A rope or hinged-link bicycle-type chain of length L and mass ρ per unit length is released from rest with x = 0. Determine the expression for the total force R exerted on the fixed platform by the chain as a function of x. Note that the hinged-link chain is a conservative system during all but
The circular sector rotates about a fixed axis through point O with angular velocity ω = 2 rad/s and angular acceleration α = 4 rad/s2 with directions as indicated in the figure. Determine the instantaneous velocity and acceleration of point A. y --x 15° 150 0.3 m 'A
The total linear momentum of a system of five particles at time t = 2.2 s is given by G2.2 = 3.4i − 2.6j + 4.6k kg∙m/s. At time t = 2.4 s, the linear momentum has changed to G2.4 = 3.7i − 2.2j + 4.9k kg∙m/s. Calculate the magnitude F of the time average of the resultant of the external
Three monkeys A, B, and C weighing 20, 25, and 15 lb, respectively, are climbing up and down the rope suspended from D. At the instant represented, A is descending the rope with an acceleration of 5 ft/sec2, and C is pulling himself up with an acceleration of 3 ft/sec2. Monkey B is climbing up with
A department-store escalator makes an angle of 30° with the horizontal and takes 40 seconds to transport a person from the first to the second floor with a vertical rise of 20 ft. At a certain instant, there are 10 people on the escalator averaging 150 lb per person and standing at rest relative
The 50,000-lb flatcar supports a 15,000-lb vehicle on a 5° ramp built on the flatcar. If the vehicle is released from rest with the flatcar also at rest, determine the velocity v of the flatcar when the vehicle has rolled s = 40 ft down the ramp just before hitting the stop at B. Neglect all
A 60-kg rocket is fired from O with an initial velocity v0 = 125 m/s along the indicated trajectory. The rocket explodes 7 seconds after launch and breaks into three pieces A, B, and C having masses of 10, 30, and 20 kg, respectively. Pieces B and C are recovered at the impact coordinates shown.
A horizontal bar of mass m1 and small diameter is suspended by two wires of length l from a carriage of mass m2 which is free to roll along the horizontal rails. If the bar and carriage are released from rest with the wires making an angle θ with the vertical, determine the velocity vb/c of the
In the unstretched position the coils of the 3-lb spring are just touching one another, as shown in part a of the figure. In the stretched position the force P, proportional to x, equals 200 lb when x = 20 in. If end A of the spring is suddenly released, determine the velocity vA of the coil end A,
The experimental race car is propelled by a rocket motor and is designed to reach a maximum speed v = 300 mi/hr under the thrust T of its motor. Prior wind-tunnel tests disclose that the wind resistance at this speed is 225 lb. If the rocket motor is burning fuel at the rate of 3.5 lb/sec,
The jet aircraft has a mass of 4.6 Mg and a drag (air resistance) of 32 kN at a speed of 1000 km/h at a particular altitude. The aircraft consumes air at the rate of 106 kg /s through its intake scoop and uses fuel at the rate of 4 kg/s. If the exhaust has a rearward velocity of 680 m/s relative to
Fresh water issues from the nozzle with a velocity of 30 m/s at the rate of 0.05 m3/s and is split into two equal streams by the fixed vane and deflected through 60° as shown. Calculate the force F required to hold the vane in place. The density of water is 1000 kg/m3. 60° А F 60,
In an unwise effort to remove debris, a homeowner directs the nozzle of his backpack blower directly toward the garage door. The nozzle velocity is 130 mi/hr and the flow rate is 410 ft3/min. Estimate the force F exerted by the airflow on the door. The specific weight of air is 0.0753 lb/ft3.
The jet water ski has reached its maximum velocity of 70 km/h when operating in salt water. The water intake is in the horizontal tunnel in the bottom of the hull, so the water enters the intake at the velocity of 70 km/h relative to the ski. The motorized pump discharges water from the horizontal
The 25-mm steel slab 1.2 m wide enters the rolls at the speed of 0.4 m /s and is reduced in thickness to 19 mm. Calculate the small horizontal thrust T on the bearings of each of the two rolls. 19 mm 25 mm 0.4 m/s
The fire tug discharges a stream of salt water (density 1030 kg/m3) with a nozzle velocity of 40 m/s at the rate of 0.080 m3/s. Calculate the propeller thrust T which must be developed by the tug to maintain a fixed position while pumping. 30°
The pump shown draws air with a density ρ through the fixed duct A of diameter d with a velocity u and discharges it at high velocity v through the two outlets B. The pressure in the airstreams at A and B is atmospheric. Determine the expression for the tension T exerted on the pump unit through
A jet-engine noise suppressor consists of a movable duct which is secured directly behind the jet exhaust by cable A and deflects the blast directly upward. During a ground test, the engine sucks in air at the rate of 43 kg/s and burns fuel at the rate of 0.8 kg/s. The exhaust velocity is 720 m/s.
The 90° vane moves to the left with a constant velocity of 10 m/s against a stream of fresh water issuing with a velocity of 20 m/s from the 25-mmdiameter nozzle. Calculate the forces Fx and Fy on the vane required to support the motion. y -20 m/s x - F, 10 m/s Fy
The pipe bend shown has a cross-sectional area A and is supported in its plane by the tension T applied to its flanges by the adjacent connecting pipes (not shown). If the velocity of the liquid is v, its density ρ, and its static pressure p, determine T and show that it is independent of the
A jet of fluid with cross-sectional area A and mass density ρ issues from the nozzle with a velocity v and impinges on the inclined trough shown in section. Some of the fluid is diverted in each of the two directions. If the trough is smooth, the velocity of both diverted streams remains v, and
The 8-oz ball is supported by the vertical stream of fresh water which issues from the 1/2-in.-diameter nozzle with a velocity of 35 ft /sec. Calculate the height h of the ball above the nozzle. Assume that the stream remains intact and there is no energy lost in the jet stream. 글 lb h /-
A jet-engine thrust reverser to reduce an aircraft speed of 200 km/h after landing employs folding vanes which deflect the exhaust gases in the direction indicated. If the engine is consuming 50 kg of air and 0.65 kg of fuel per second, calculate the braking thrust as a fraction n of the engine
Water issues from a nozzle with an initial velocity v and supports a thin plate of mass m at a height h above the nozzle exit. A hole in the center of the plate allows some of the water to travel upward to a maximum altitude 2h above the plate. Determine the mass m of the plate. Neglect any effects
The axial-flow fan C pumps air through the duct of circular cross section and exhausts it with a velocity v at B. The air densities at A and B are ρA and ρB, respectively, and the corresponding pressures are pA and pB. The fixed deflecting blades at D restore axial flow to the air after it
Air is pumped through the stationary duct A with a velocity of 50 ft/sec and exhausted through an experimental nozzle section BC. The average static pressure across section B is 150 lb/in.2 gage, and the specific weight of air at this pressure and at the temperature prevailing is 0.840 lb/ft3. The
Air enters the pipe at A at the rate of 6 kg/s under a pressure of 1400 kPa gage and leaves the whistle at atmospheric pressure through the opening at B. The entering velocity of the air at A is 45 m/s, and the exhaust velocity at B is 360 m/s. Calculate the tension T, shear V, and bending moment M
The sump pump has a net mass of 310 kg and pumps fresh water against a 6-m head at the rate of 0.125 m3/s. Determine the vertical force R between the supporting base and the pump flange at A during operation. The mass of water in the pump may be taken as the equivalent of a 200-mm-diameter column 6

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