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engineering
engineering mechanics dynamics

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

The solid right-circular cone of base radius r and height h rolls on a flat surface without slipping. The center B of the circular base moves in a circular path around the z-axis with a constant speed v. Determine the angular velocity w and the angular acceleration a of the solid cone. B, h A x- - y
The solid cylinder has a body cone with a semi-vertex angle of 20°. Momentarily the angular velocity w has a magnitude of 30 rad/s and lies in the y-z plane. Determine the rate p at which the cylinder is spinning about its z-axis and write the vector expression for the velocity of B with respect
The helicopter is nosing over at the constant rate q rad/s. If the rotor blades revolve at the constant speed p rad/s, write the expression for the angular acceleration a of the rotor. Take the y-axis to be attached to the fuselage and pointing forward perpendicular to the rotor axis.
The collar at O and attached shaft OC rotate about the fixed x0-axis at the constant rate Ω = 4 rad/s. Simultaneously, the circular disk rotates about OC at the constant rate p = 10 rad/s. Determine the magnitude of the total angular velocity w of the disk and fi nd its angular acceleration a. y
If the angular rate p of the disk in Prob. 7/31 is increasing at the rate of 6 rad/s per second and if Ω remains constant at 4 rad/s, determine the angular acceleration a of the disk at the instant when p reaches 10 rad/s. y 400 mm A 300 mm 2 = 4 rad/s p = 10 rad/s Problem 7/31
For the conditions of Prob. 7/31, determine the velocity vA and acceleration aA of point A on the disk as it passes the position shown. Reference axes x-y-z are attached to the collar at O and its shaft OC. y 400 mm A 300 mm 2 = 4 rad/s p = 10 rad/s Problem 7/31
An unmanned radar-radio controlled aircraft with tilt-rotor propulsion is being designed for reconnaissance purposes. Vertical rise begins with θ = 0 and is followed by horizontal flight as θ approaches 90°. If the rotors turn at a constant speed N of 360 rev/min, determine the angular
End A of the rigid link is confined to move in the −x-direction while end B is confined to move along the z-axis. Determine the component wn normal to AB of the angular velocity of the link as it passes the position shown with vA = 3 ft/sec. Be 7' VA 2' 3'- |-
The small motor M is pivoted about the x-axis through O and gives its shaft OA a constant speed p rad/s in the direction shown relative to its housing. The entire unit is then set into rotation about the vertical Z-axis at the constant angular velocity Ω rad/s. Simultaneously, the motor pivots
The flight simulator is mounted on six hydraulic actuators connected in pairs to their attachment points on the underside of the simulator. By programming the actions of the actuators, a variety of flight conditions can be simulated with translational and rotational displacements through a limited
The robot of Prob. 7/16 is shown again here, where the coordinate system x-y-z with origin at O2 rotates about the X-axis at the rate θ˙. Nonrotating axes X-Y-Z oriented as shown have their origin at O1. If w2 = θ˙ = 3 rad/s constant, w3 = 1.5 rad/s constant, w1 = w5 = 0, O1O2 = 1.2 m, and
For the instant represented collar B is moving along the fixed shaft in the X-direction with a constant velocity vB = 4 m/s. Also at this instant X = 0.3 m and Y = 0.2 m. Calculate the velocity of collar A, which moves along the fixed shaft parallel to the Y-axis. Solve, first, by differentiating
The spacecraft is revolving about its z-axis, which has a fixed space orientation, at the constant rate p = 1/10 rad/s. Simultaneously, its solar panels are unfolding at the rate β˙ which is programmed to vary with β as shown in the graph. Determine the angular acceleration a of panel A an
The disk has a constant angular velocity p about its z-axis, and the yoke A has a constant angular velocity w2 about its shaft as shown. Simultaneously, the entire assembly revolves about the fixed X-axis with a constant angular velocity w1. Determine the expression for the angular acceleration of
The collar and clevis A are given a constant upward velocity of 8 in./sec for an interval of motion and cause the ball end of the bar to slide in the radial slot in the rotating disk. Determine the angular acceleration of the bar when the bar passes the position for which z = 3 in. The disk turns
The circular disk of 100-mm radius rotates about its z-axis at the constant speed p = 240 rev/min, and arm OCB rotates about the Y-axis at the constant speed N = 30 rev/min. Determine the velocity v and acceleration a of point A on the disk as it passes the position shown. Use reference axes x-y-z
Solve Prob. 7/43 by attaching the reference axes x-y-z to the rotating disk. 180 mm p -100 mm B D Y 100 mm Problem 7/43
For the conditions described in Prob. 7/36, determine the velocity v and acceleration a of the center A of the ball tool in terms of β. R. M/ Problem 7/36
The circular disk is spinning about its own axis (y-axis) at the constant rate p = 10π rad/s. Simultaneously, the frame is rotating about the Z-axis at the constant rate Ω = 4π rad/s. Calculate the angular acceleration a of the disk and the acceleration of point A at the top of the disk. Axes
The center O of the spacecraft is moving through space with a constant velocity. During the period of motion prior to stabilization, the spacecraft has a constant rotational rate Ω = 0.5 rad/sec about its z-axis. The x-y-z axes are attached to the body of the craft, and the solar panels rotate
The thin circular disk of mass m and radius r is rotating about its z-axis with a constant angular velocity p, and the yoke in which it is mounted rotates about the x-axis through OB with a constant angular velocity w1. Simultaneously, the entire assembly rotates about the fixed Y-axis through O
For the conditions specified with Sample Problem 7/2, except that γ is increasing at the steady rate of 3π rad/sec, determine the angular velocity ω and the angular acceleration α of the rotor when the position γ = 30° is passed. N - y •P A of 8" 4" 12" Problem 7/3 -
The wheel of radius r is free to rotate about the bent axle CO which turns about the vertical axis at the constant rate p rad/s. If the wheel rolls without slipping on the horizontal circle of radius R, determine the expressions for the angular velocity w and angular acceleration a of the wheel.
The gyro rotor shown is spinning at the constant rate of 100 rev/min relative to the x-y-z axes in the direction indicated. If the angle γ between the gimbal ring and the horizontal X-Y plane is made to increase at the constant rate of 4 rad/s and if the unit is forced to precess about the
For a short interval of motion, collar A moves along its fixed shaft with a velocity vA = 2 m/s in the Y-direction. Collar B, in turn, slides along its fixed vertical shaft. Link AB is 700 mm in length and can turn within the clevis at A to allow for the angular change between the clevises. For the
The three small spheres, each of mass m, are rigidly mounted to the horizontal shaft which rotates with the angular velocity w as shown. Neglect the radius of each sphere compared with the other dimensions and write expressions for the magnitudes of their linear momentum G and their angular
The spheres of Prob. 7/53 are replaced by three rods, each of mass m and length l, mounted at their centers to the shaft, which rotates with the angular velocity w as shown. The axes of the rods are, respectively, in the x-, y-, and z-directions, and their diameters are negligible compared with the
The aircraft landing gear viewed from the front is being retracted immediately after takeoff, and the wheel is spinning at the rate corresponding to the takeoff speed of 200 km/h. The 45-kg wheel has a radius of gyration about its z-axis of 370 mm. Neglect the thickness of the wheel and calculate
The bent rod has a mass ρ per unit length and rotates about the z-axis with an angular velocity w. Determine the angular momentum HO of the rod about the fixed origin O of the axes, which are attached to the rod. Also find the kinetic energy T of the rod. 9.
Use the results of Prob. 7/56 and determine the angular momentum HG of the bent rod of that problem about its mass center G using the given reference axes. b 9. Problem 7/56
The slender rod of mass m and length l rotates about the y-axis as the element of a right-circular cone. If the angular velocity about the y-axis is w, determine the expression for the angular momentum of the rod with respect to the x-y-z axes for the particular position shown.
The solid half-circular cylinder of mass m revolves about the z-axis with an angular velocity w as shown. Determine its angular momentum H with respect to the x-y-z axes. b. x-
The elements of a reaction-wheel attitude-control system for a spacecraft are shown in the figure. Point G is the center of mass for the system of the spacecraft and wheels, and x, y, z are principal axes for the system. Each wheel has a mass m and a moment of inertia I about its own axis and spins
The gyro rotor is spinning at the constant rate p = 100 rev/min relative to the x-y-z axes in the direction indicated. If the angle γ between the gimbal ring and horizontal X-Y plane is made to increase at the rate of 4 rad /sec and if the unit is forced to precess about the vertical at the
The slender steel rod AB weighs 6.20 lb and is secured to the rotating shaft by the rod OG and its fittings at O and G. The angle β remains constant at 30°, and the entire rigid assembly rotates about the z-axis at the steady rate N = 600 rev/min. Calculate the angular momentum HO of AB and its
The rectangular plate, with a mass of 3 kg and a uniform small thickness, is welded at the 45° angle to the vertical shaft, which rotates with the angular velocity of 20π rad/s. Determine the angular momentum H of the plate about O and find the kinetic energy of the plate. 100 100 mm mm 45° y w
The circular disk of mass m and radius r is mounted on the vertical shaft with an angle a between its plane and the plane of rotation of the shaft. Determine an expression for the angular momentum H of the disk about O. Find the angle β which the angular momentum H makes with the shaft if a =
The right-circular cone of height h and base radius r spins about its axis of symmetry with an angular rate p. Simultaneously, the entire cone revolves about the x-axis with angular rate Ω. Determine the angular momentum HO of the cone about the origin O of the x-y-z axes and the kinetic energy T
Each of the slender rods of length l and mass m is welded to the circular disk which rotates about the vertical z-axis with an angular velocity w. Each rod makes an angle β with the vertical and lies in a plane parallel to the y-z plane. Determine an expression for the angular momentum HO of the
The spacecraft shown has a mass m with mass center G. Its radius of gyration about its z-axis of rotational symmetry is k and that about either the x- or y-axis is k′. In space, the spacecraft spins within its x-y-z reference frame at the rate p = Φ˙. Simultaneously, a point C on the z-axis
The uniform circular disk of Prob. 7/48 with the three components of angular velocity is shown again here. Determine the kinetic energy T and the angular momentum HO with respect to O of the disk for the instant represented, when the x-y plane coincides with the X-Y plane. The mass of the disk is
The 4-in.-radius wheel weighs 6 lb and turns about its y′-axis with an angular velocity p = 40π rad/sec in the direction shown. Simultaneously, the fork rotates about its x-axis shaft with an angular velocity w = 10π rad/sec as indicated. Calculate the angular momentum of the wheel about its
The assembly, consisting of the solid sphere of mass m and the uniform rod of length 2c and equal mass m, revolves about the vertical z-axis with an angular velocity w. The rod of length 2c has a diameter which is small compared with its length and is perpendicular to the horizontal rod to which it
In a test of the solar panels for a spacecraft, the model shown is rotated about the vertical axis at the angular rate w. If the mass per unit area of panel is p, write the expression for the angular momentum HO of the assembly about the axes shown in terms of θ. Also determine the maximum,
Each of the two rods of mass m is welded to the face of the disk, which rotates about the vertical axis with a constant angular velocity w. Determine the bending moment M acting on each rod at its base.
The slender shaft carries two offset particles, each of mass m, and rotates about the z-axis with the constant angular rate w as indicated. Determine the x- and y-components of the bearing reactions at A and B due to the dynamic imbalance of the shaft for the position shown. R т B /3
The uniform slender bar of length l and mass m is welded to the shaft, which rotates in bearings A and B with a constant angular velocity w. Determine the expression for the force supported by the bearing at B as a function of θ. Consider only the force due to the dynamic imbalance and assume that
If a torque M = Mk is applied to the shaft in Prob. 7/75, determine the x- and y-components of the force supported by the bearing B as the bar and shaft start from rest in the position shown. Neglect the mass of the shaft and consider dynamic forces only. A B b- Problem 7/75
The paint stirrer shown in the figure is made from a rod of length 7b and mass p per unit length. Before immersion in the paint, the stirrer is rotating freely at a constant high angular velocity w about its z-axis. Determine the bending moment M in the rod at the base O of the chuck. to
The 6-kg circular disk and attached shaft rotate at a constant speed w = 10 000 rev/min. If the center of mass of the disk is 0.05 mm off center, determine the magnitudes of the horizontal forces A and B supported by the bearings because of the rotational imbalance. 150 mm A 200 mm B
Determine the bending moment M at the tangency point A in the semicircular rod of radius r and mass m as it rotates about the tangent axis with a constant and large angular velocity w. Neglect the moment mgr produced by the weight of the rod. A
If the semicircular rod of Prob. 7/79 starts from rest under the action of a torque MO applied through the collar about its z-axis of rotation, determine the initial bending moment M in the rod at A. A Problem 7/79
The large satellite-tracking antenna has a moment of inertia I about its z-axis of symmetry and a moment of inertia IO about each of the x- and y-axes. Determine the angular acceleration a of the antenna about the vertical Z-axis caused by a torque M applied about Z by the drive mechanism for a
The plate has a mass of 3 kg and is welded to the fixed vertical shaft, which rotates at the constant speed of 20π rad/s. Compute the moment M applied to the shaft by the plate due to dynamic imbalance. 100 , 100 mm/ m m 45° -y o = 20n rad/s -200 mm -200 mm
Each of the two semicircular disks has a mass of 1.20 kg and is welded to the shaft supported in bearings A and B as shown. Calculate the forces applied to the shaft by the bearings for a constant angular speed N = 1200 rev/min. Neglect the forces of static equilibrium. C 80 mm/80 mm /80 mm 80 mm
Solve Prob. 7/83 for the case where the assembly starts from rest with an initial angular acceleration a = 900 rad/s2 as a result of a starting torque (couple) M applied to the shaft in the same sense as N. Neglect the moment of inertia of the shaft about its z-axis and calculate M. C 80 mm 80 mm
The uniform slender bar of mass p per unit length is freely pivoted about the y-axis at the clevis, which rotates about the fixed vertical z-axis with a constant angular velocity w. Determine the steady-state angle θ assumed by the bar. Length b is greater than length c. -y
The circular disk of mass m and radius r is mounted on the vertical shaft with a small angle a between its plane and the plane of rotation of the shaft. Determine the expression for the bending moment M acting on the shaft due to the wobble of the disk at a shaft speed of w rad/s. - x x' y
Determine the normal forces under the two disks of Sample Problem 7/7 for the position where the plane of the curved bar is vertical. Take the curved bar to be at the top of disk A and at the bottom of disk B. N2 - B N1 A Problem 7/7
For the plate of mass m in Prob. 7/89, determine the y- and z-components of the moment applied to the plate by the weld at O necessary to give the plate an angular acceleration a = w˙ starting from rest. Neglect the moment due to the weight. b/2 b/2 Problem 7/89
If the homogeneous triangular plate of Prob. 7/93 is released from rest in the position shown, determine the magnitude of the bearing reaction at A after the plate has rotated 90°. y B A Problem 7/93
If the wheel in case (a) of Prob. 7/108 is forced to precess about the vertical by a mechanical drive at the steady rate Ω = 2j rad/s, determine the bending moment in the horizontal shaft at A. In the absence of friction, what torque MO is applied to the collar at O to sustain this motion? y y 400
The earth-scanning satellite is in a circular orbit of period τ. The angular velocity of the satellite about its y- or pitch-axis is w = 2π/τ, and the angular rates about the x- and z-axes are zero. Thus, the x-axis of the satellite always points to the center of the earth. The satellite has a
The solid cone of mass m, base radius r, and altitude h is spinning at a high rate p about its own axis and is released with its vertex O supported by a horizontal surface. Friction is sufficient to prevent the vertex from slipping in the x-y plane. Determine the direction of the precession Ω and
The circular disk of radius r is mounted on its shaft which is pivoted at O so that it may rotate about the vertical z0-axis. If the disk rolls at constant speed without slipping and makes one complete turn around the circle of radius R in time τ, determine the expression for the absolute angular
Determine the angular acceleration a for the rolling circular disk of Prob. 7/135. Use the results cited in the answer for that problem. A R- Problem 7/135
Determine the velocity v of point A on the disk of Prob. 7/135 for the position shown. A R- Problem 7/135
Determine the acceleration a of point A on the disk of Prob. 7/135 for the position shown. R- Problem 7/135
Rework Prob. 7/140 if β, instead of being constant at 20°, is increasing at the steady rate of 120 rev/min. Find the angular momentum HO of the disk for the instant when β = 20°. Also compute the kinetic energy T of the disk. Is T dependent on β? N -y 10" 4" Problem 7/140
Calculate the bending moment M in the shaft at O for the rotating assembly of Prob. 7/144 as it starts from rest with an initial angular acceleration of 200 rad/s2. N - 150 mm 75 mm 150 mm 75 mm Problem 7/144
For the system of Prob. 8/2, determine the position x of the mass as a function of time if the mass is released from rest at time t = 0 from a position 2 inches to the left of the equilibrium position. Determine the maximum velocity and maximum acceleration of the mass over one cycle of motion. k =
For the spring-mass system shown, determine the static deflection δst, the system period τ, and the maximum velocity vmax which result if the cylinder is displaced 100 mm downward from its equilibrium position and released from rest. k = 98 N/m Equilibrium position m = 2 kg y
The cylinder of the system of Prob. 8/5 is displaced 100 mm downward from its equilibrium position and released at time t = 0. Determine the position y, velocity v, and acceleration a when t = 3 s. What is the maximum acceleration? k = 98 N/m Equilibrium position m = 2 kg y Problem 8/5
For the cylinder of Prob. 8/10, determine the vertical displacement x, measured positive down in millimeters from the equilibrium position, in terms of the time t in seconds measured from the instant of release from the position of 25 mm added deflection. 30 kg Problem 8/10
The 8-lb body of Prob. 8/25 is released from rest a distance x0 to the right of the equilibrium position. Determine the displacement x as a function of time t, where t = 0 is the time of release. c = 2.5 lb-sec/ft 8 lb www k = 3 lb/in. Problem 8/25
The owner of a 3400-lb pickup truck tests the action of his rear-wheel shock absorbers by applying a steady 100-lb force to the rear bumper and measuring a static deflection of 3 in. Upon sudden release of the force, the bumper rises and then falls to a maximum of 12 in. below the unloaded
The 2-kg mass is released from rest at a distance x0 to the right of the equilibrium position. Determine the displacement x as a function of time. c = 42 N-s/m 2 kg w k = 98 N/m
Develop the equation of motion in terms of the variable x for the system shown. Determine an expression for the damping ratio ζ in terms of the given system properties. Neglect the mass of the crank AB and assume small oscillations about the equilibrium position shown. OA a т В
Investigate the case of Coulomb damping for the block shown, where the coefficient of kinetic friction is μk and each spring has a stiffness k/2. The block is displaced a distance x0 from the neutral position and released. Determine and solve the differential equation of motion. Plot the resulting
A viscously damped spring-mass system is excited by a harmonic force of constant amplitude F0 but varying frequency w. If the amplitude of the steady-state motion is observed to decrease by a factor of 8 as the frequency ratio w/wn is varied from 1 to 2, determine the damping ratio ζ of the system.
The 30-kg cart is acted upon by the harmonic force shown in the figure. If c = 0, determine the range of the driving frequency w for which the magnitude of the steady-state response is less than 75 mm. k = 1080 N/m www 30 kg F = 25 cos wt N
If the viscous damping coefficient of the damper in the system of Prob. 8/47 is c = 36 N ∙ s/m, determine the range of the driving frequency w for which the magnitude of the steady-state response is less than 75 mm. k = 1080 N/m www 30 kg ►F = 25 cos ot N Problem 8/47
If the driving frequency for the system of Prob. 8/47 is w = 6 rad/s, determine the required value of the damping coefficient c if the steady-state amplitude is not to exceed 75 mm. k = 1080 N/m www 30 kg F = 25 cos ot N Problem 8/47
Derive and solve the equation of motion for the mass m in terms of the variable x for the system shown. Neglect the mass of the lever AOC and assume small oscillations. C k2 k1 ww т TxB = bo cos ot A a
The seismic instrument is mounted on a structure which has a vertical vibration with a frequency of 5 Hz and a double amplitude of 18 mm. The sensing element has a mass m = 2 kg, and the spring stiffness is k = 1.5 kN/m. The motion of the mass relative to the instrument base is recorded on a
Derive the expression for the energy loss E over a complete steady-state cycle due to the frictional dissipation of energy in a viscously damped linear oscillator. The forcing function is F0 sin wt, and the displacement-time relation for steady-state motion is xP = X sin (wt − Φ) where the
Determine the amplitude of vertical vibration of the car as it travels at a velocity v = 40 km/h over the wavy road whose contour may be expressed as a sine or cosine function with a double amplitude 2b = 50 mm. The mass of the car is 1800 kg and the stiffness of each of the four car springs is 35
The light rod and attached small spheres of mass m each are shown in the equilibrium position, where all four springs are equally precompressed. Determine the natural frequency wn and period τ for small oscillations about the frictionless pivot O. L k k m т k k
A uniform rectangular plate pivots about a horizontal axis through one of its corners as shown. Determine the natural frequency wn of small oscillations. a 9.
The thin square plate is suspended from a socket (not shown) which fits the small ball attachment at O. If the plate is made to swing about axis A-A, determine the period for small oscillations. Neglect the small offset, mass, and friction of the ball. B A b/2 b/2 А B
If the square plate of Prob. 8/73 is made to oscillate about axis B-B, determine the period of small oscillations. B A b/2 b/2 -A B Problem 8/73
The uniform rod of length l and mass m is suspended at its midpoint by a wire of length L. The resistance of the wire to torsion is proportional to its angle of twist θ and equals (JG/L)θ where J is the polar moment of inertia of the wire cross section and G is the shear modulus of elasticity.
The uniform sector has mass m and is freely hinged about a horizontal axis through point O. Determine the equation of motion of the sector for large-amplitude vibrations about the equilibrium position. State the period τ for small oscillations about the equilibrium position if r = 325 mm and
The assembly of mass m is formed from uniform and slender welded rods and is freely hinged about a horizontal axis through O. Determine the equation of motion of the assembly for large-amplitude vibrations about the equilibrium position. State the period τ for small oscillations about the
The thin-walled cylindrical shell of radius r and height h is welded to the small shaft at its upper end as shown. Determine the natural circular frequency wn for small oscillations of the shell about the y-axis. r- Jー h
Determine the system equation of motion in terms of the variable shown in the figure. Assume small angular motion of bar OA, and neglect the mass of link CD. 2L 3 - Xg = b cos ot Co ww. D k L m2 3 m1 A
The uniform rod of mass m is freely pivoted about a horizontal axis through point O. Assume small oscillations and determine an expression for the damping ratio ζ. For what value ccr of the damping coefficient c will the system be critically damped? k
The triangular frame of mass m is formed from uniform slender rod and is suspended from a socket (not shown) which fi ts the small ball attachment at O. If the frame is made to swing about axis A-A, determine the natural circular frequency wn for small oscillations. Neglect the small offset, mass,
The uniform rod of mass m is freely pivoted about point O. Assume small oscillations and determine an expression for the damping ratio ζ. For what value ccr of the damping coefficient c will the system be critically damped? a k
The mechanism shown oscillates in the vertical plane about the pivot O. The springs of equal stiffness k are both compressed in the equilibrium position θ = 0. Determine an expression for the period τ of small oscillations about O. The mechanism has a mass m with mass center G, and the

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