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engineering
engineering mechanics dynamics

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

When the motor is slowly brought up to speed, a rather large vibratory oscillation of the entire motor about O-O occurs at a speed of 360 rev/min, which shows that this speed corresponds to the natural frequency of free oscillation of the motor. If the motor has a mass of 43 kg and radius of
The system of Prob. 8/35 is repeated here with the added information that link AOB now has mass m3 and radius of gyration kO about point O. Ignore friction and derive the differential equation of motion for the system shown in terms of the variable x1. >X1 k1 www C1 A m1 X2 k2 m3, ko m2 B C2
Determine the value meff of the mass of system (b) so that the frequency of system (b) is equal to that of system (a). Note that the two springs are identical and that the wheel of system (a) is a solid homogeneous cylinder of mass m2. The cord does not slip on the cylinder. k m2 ww meff m1 (a) (b)
The system of Prob. 8/43 is repeated here. If the crank AB now has mass m2 and a radius of gyration kO about point O, determine expressions for the undamped natural frequency wn and the damping ratio ζ in terms of the given system properties. Assume small oscillations. The damping coefficient for
The two masses are connected by an inextensible cable which passes securely over the periphery of the cylindrical pulley of mass m3, radius r, and radius of gyration kO. Determine the equation of motion for the system in terms of the variable x. State the critical driving frequency wc of the
The lower spring of Prob. 8/91 is replaced by a damper. Determine the equation of motion for the system in terms of the variable x. State the value of the viscous damping coefficient c which will give a damping ratio ζ = 0.2. Evaluate for m1 = 25 kg, m2 = 10 kg, m3 = 15 kg, k = 450 N/m, r = 300
The uniform solid cylinder of mass m and radius r rolls without slipping during its oscillation on the circular surface of radius R. If the motion is confined to small amplitudes θ = θ0, determine the period τ of the oscillations. Also determine the angular velocity w of the cylinder as it
The assembly of Prob. 8/40 is repeated here with the additional information that body ABC now has mass m4 and a radius of gyration kO about its pivot at O, about which it is balanced. If a harmonic torque M = M0 cos wt is applied to body ABC, determine the equation of motion for the system in terms
The elements of the “swing-axle” type of independent rear suspension for automobiles are depicted in the figure. The differential D is rigidly attached to the car frame. The half-axles are pivoted at their inboard ends (point O for the half-axle shown) and are rigidly attached to the wheels.
The 1.5-kg bar OA is suspended vertically from the bearing O and is constrained by the two springs each of stiffness k = 120 N/m and both equally precompressed with the bar in the vertical equilibrium position. Treat the bar as a uniform slender rod and compute the natural frequency ƒn of small
The light rod and attached sphere of mass m are at rest in the horizontal position shown. Determine the period τ for small oscillations in the vertical plane about the pivot O. m ー6一+ー6 -b- -
A uniform rod of mass m and length l is welded at one end to the rim of a light circular hoop of radius l. The other end lies at the center of the hoop. Determine the period τ for small oscillations about the vertical position of the bar if the hoop rolls on the horizontal surface without
The spoked wheel of radius r, mass m, and centroidal radius of gyration k̅ rolls without slipping on the incline. Determine the natural frequency of oscillation and explore the limiting cases of k̅ = 0 and k̅ = r. k wwww m
Determine the period τ for the uniform circular hoop of radius r as it oscillates with small amplitude about the horizontal knife edge.Solve the following problems by the energy method.
The length of the spring is adjusted so that the equilibrium position of the arm is horizontal as shown. Neglect the mass of the spring and the arm and calculate the natural frequency ƒn for small oscillations. m k ewww
The quarter-circular sector of mass m and radius r is set into small rocking oscillation on the horizontal surface. If no slipping occurs, determine the expression for the period τ of each complete oscillation.
The homogeneous circular cylinder of Prob. 8/93, repeated here, rolls without slipping on the track of radius R. Determine the period τ for small oscillations. R G
The disk has mass moment of inertia IO about O and is acted upon by a torsional spring of constant kT. The position of the small sliders, each of which has mass m, is adjustable. Determine the value of x for which the system has a given period τ. Io X. т m kT
The assembly shown consists of two sheaves of mass m1 = 35 kg and m2 = 15 kg, outer groove radii r1 = 525 mm and r2 = 250 mm, and centroidal radii of gyration (kO)1 = 350 mm and (kO)2 = 150 mm. The sheaves are fitted to a central shaft at O with bearings which allow them to rotate independently of
The semicircular cylindrical shell of radius r with small but uniform wall thickness is set into small rocking oscillation on the horizontal surface. If no slipping occurs, determine the expression for the period τ of each complete oscillation.
A hole of radius R/4 is drilled through a cylinder of radius R to form a body of mass m as shown. If the body rolls on the horizontal surface without slipping, determine the period τ for small oscillations. RI4 R/2 R/4 of
The body consists of two slender uniform rods which have a mass per unit length. The rods are welded together and pivot about a horizontal axis through O against the action of a torsional spring of stiffness kT. By the method of this article, determine the natural circular frequency wn for
By the method of this article, determine the period of vertical oscillation. Each spring has a stiffness of 6 lb/in., and the mass of the pulleys may be neglected. 50 lb
The uniform slender rod of length l and mass m2 is secured to the uniform disk of radius l/5 and mass m1. If the system is shown in its equilibrium position, determine the natural frequency wn and the maximum angular velocity w for small oscillations of amplitude θ0 about the pivot O. m1 m2 21/5
Derive the natural frequency ƒn of the system composed of two homogeneous circular cylinders, each of mass M, and the connecting link AB of mass m. Assume small oscillations. M M ro B, To m LA
The rotational axis of the turntable is inclined at an angle a from the vertical. The turntable shaft pivots freely in bearings which are not shown. If a small block of mass m is placed a distance r from point O, determine the natural frequency wn for small rotational oscillations through the angle
The assembly of Prob. 8/95 is repeated here without the applied harmonic torque. By the method of this article, determine the equation of motion of the system in terms of x for small oscillations about the equilibrium configuration if c1 = c2 = c3 = 0. C3 k3 m3 B a m4, ko m1 k1 C1 k2 m2 B. C2
The ends of the uniform bar of mass m slide freely in the vertical and horizontal slots as shown. If the bar is in static equilibrium when = 0, determine the natural frequency wn of small oscillations. What condition must be imposed on the spring constant k in order that oscillations take place? k k
The 12-kg block is supported by the two 5-kg links with two torsion springs, each of constant kT = 500 N ∙ m/rad, arranged as shown. The springs are sufficiently stiff so that stable equilibrium is established in the position shown. Determine the natural frequency ƒn for small oscillations about
The front-end suspension of an automobile is shown. Each of the coil springs has a stiffness of 270 lb/in. If the weight of the front-end frame and equivalent portion of the body attached to the front end is 1800 lb, determine the natural frequency ƒn of vertical oscillation of the frame and body
Determine the natural frequency ƒn of the inverted pendulum. Assume small oscillations, and note any restrictions on your solution. m k Aww k
Determine the period τ of small oscillations for the uniform sector of mass m = 1.0 kg, side b = 675 mm and angle β = 18°. The torsional spring of modulus kT = 8 N · m/rad is undeflected when the sector is in the position shown. kT b m
The 0.1-kg projectile is fi red into the 10-kg block which is initially at rest with no force in the spring. The spring is attached at both ends. Calculate the maximum horizontal displacement X of the spring and the ensuing period of oscillation of the block and embedded projectile. k = 3 kN/m 0.1
The uniform circular disk is suspended by a socket (not shown) which fits over the small ball attachment at O. Determine the frequency of small motion if the disk swings freely about (a) Axis A-A (b) Axis B-B. Neglect the small offset, mass, and friction of the ball. A B B
Determine the natural frequency ƒn for small oscillations of the semicircular slender rod about the equilibrium position if (a) r = 150 mm.(b) r = 300 mm.(c) r = 600 mm. Motion takes place in a vertical plane and friction in the pivot at O is negligible.
The triangular frame is constructed of uniform slender rod and pivots about a horizontal axis through point O. Determine the critical driving frequency wc of the block B which will result in excessively large oscillations of the assembly. The total mass of the frame is m. SB = b sin ot d B k d
A linear oscillator with mass m, spring constant k, and viscous damping coefficient c is set into motion when released from a displaced position. Derive an expression for the energy loss Q during one complete cycle in terms of the amplitude x1 at the start of the cycle.
Calculate the damping ratio ζ of the system shown if the weight and radius of gyration of the stepped cylinder are W = 20 lb and k̅ = 5.5 in., the spring constant is k = 15 lb/in., and the damping coefficient of the hydraulic cylinder is c = 2 lb-sec/ft. The cylinder rolls without slipping on the
Determine the value of the viscous damping coefficient c for which the system is critically damped. The cylinder mass is m = 2 kg and the spring constant is k = 150 N/m. Neglect the mass and friction of the pulley. m
The seismic instrument shown is secured to a ship’s deck near the stern where propeller-induced vibration is most pronounced. The ship has a single three-bladed propeller which turns at 180 rev/min and operates partly out of water, thus causing a shock as each blade breaks the surface. The
The square plate of side length 2b pivots about its center point O against the action of four springs, each of stiffness k. Each spring is attached to the corner of the plate at one end and to the center of a slotted collar at the other end. As the plate rotates, a rod slides through the smooth
An experimental engine weighing 480 lb is mounted on a test stand with spring mounts at A and B, each with a stiffness of 600 lb/in. The radius of gyration of the engine about its mass center G is 4.60 in. With the motor not running, calculate the natural frequency (ƒn)y of vertical vibration and
The uniform bar of mass M and length l has a small roller of mass m with negligible bearing friction at each end. Determine the period ???? of the system for small oscillations on the curved track. The radius of gyration of the rollers is negligible. R m т M.
A 200-kg machine rests on four floor mounts, each of which has an effective spring constant k = 250 kN/m and an effective viscous damping coefficient c = 1000 N∙ s/m. The floor is known to vibrate vertically with a frequency of 24 Hz. What would be the effect on the amplitude of the absolute
The mass of a critically damped system having a natural frequency wn = 4 rad/s is released from rest at an initial displacement x0. Determine the time t required for the mass to reach the position x = 0.1x0.
The uniform sector of Prob. 8/77 is repeated here with m = 4 kg, r = 325 mm, and β = 45°. If the sector is released from rest with θ0 = 90°, plot the value of for the time period 0 ≤ t ≤ 6 s. Friction in the pivot at O results in a resistive torque of magnitude M = cθ˙, where the
The mass of the system shown is released with the initial conditions x0 = 0.1 m and x˙0 = −5 m/s at t = 0. Plot the response of the system and determine the time(s) (if any) at which the displacement x = −0.05 m. 100 N/m 2 kg 50 N.s/m
Shown in the figure are the elements of a displacement meter used to study the motion yB = b sin wt of the base. The motion of the mass relative to the frame is recorded on the rotating drum. If l1 = 1.2 ft, l2 = 1.6 ft, l3 = 2 ft, W = 2 lb, c = 0.1 lb-sec/ft, and w = 10 rad/sec, determine the
The 4-kg mass is suspended by the spring of stiffness k = 350 N/m and is initially at rest in the equilibrium position. If a downward force F = Ct is applied to the body and reaches a value of 40 N when t = 1 s, derive the differential equation of motion, obtain its solution, and plot the
Plot the response x of the 50-lb body over the time interval 0 ≤ t ≤ 1 second. Determine the maximum and minimum values of x and their respective times. The initial conditions are x0 = 0 and x˙0 = 6 ft/sec. |F = (160 cos 60t) lb 100 lb/in. 18 lb-sec/ft www 50 lb
Determine and plot the response x as a function of time for the undamped linear oscillator subjected to the force F which varies linearly with time for the first 3/4 second as shown. The mass is initially at rest with x = 0 at time t = 0. F, N 6.25 H F k = 90 N/m www 0.75 kg - t,s 3/4
The 4-kg cylinder is attached to a viscous damper and to the spring of stiffness k = 800 N/m. If the cylinder is released from rest at time t = 0 from the position where it is displaced a distance y = 100 mm from its equilibrium position, plot the displacement y as a function of time for the first
Determine the mass moment of inertia of the bent uniform slender rod about the x- and y-axes shown, and about the z-axis. The rod has a mass ρ per unit length. y 2. /2
Determine the mass moment of inertia of the uniform thin triangular plate of mass m about the x-axis. Also determine the radius of gyration about the x-axis. By analogy state Iyy and ky. Then determine Izz and kz. y m h
Determine the mass moment of inertia about the y-axis for the uniform thin equilateral triangular plate of mass m. Also determine its radius of gyration about the y-axis. y b. 21
Calculate the mass moment of inertia of the homogeneous right-circular cone of mass m, base radius r, and altitude h about the cone axis x and about the y-axis through its vertex. y
Determine the mass moment of inertia of the uniform thin parabolic plate of mass m about the x-axis. State the corresponding radius of gyration. y 2 -Parabolic h m
For the thin homogeneous plate of uniform thickness t and mass m, determine the mass moments of inertia about the x'-, y'-, and z'-axes through the end of the plate at A. Refer to the results of Sample Problem B/4 and Table D /3 in Appendix D as needed. y y' x = ky? b- x, x'
Determine the mass moment of inertia about the x-axis of the thin elliptical plate of mass m. y by a
Determine the mass moment of inertia of the homogeneous solid of revolution of mass m about the x-axis. y y = kxl.5 m
Determine the mass moment of inertia of the homogeneous solid of revolution of the previous problem about the y- and z-axes.
Determine the radius of gyration about the y-axis for the steel part shown in section. The part is formed by revolving one of the trapezoidal areas 360° around the y-axis. y -30 -30- 40 - -x 20- 10 Dimensions in millimeters
Develop an expression for the mass moment of inertia of the homogeneous solid of revolution of mass m about the y-axis. a y = kzn b. m
Determine the mass moment of inertia about the x-axis of the solid spherical segment of mass m. y R 2. -x-
Determine the moment of inertia about the generating axis of a complete ring (torus) of mass m having a circular section with the dimensions shown in the sectional view. -R-
The plane area shown in the top portion of the figure is rotated 180° about the x-axis to form the body of revolution of mass m shown in the lower portion of the figure. Determine the mass moment of inertia of the body about the x-axis. y Parabolic 26 26 L 2 2
Determine Iyy for the homogeneous body of revolution of the previous problem.
The thickness of the homogeneous triangular plate of mass m varies linearly with the distance from the vertex toward the base. The thickness a at the base is small compared with the other dimensions. Determine the moment of inertia of the plate about the y-axis along the centerline of the base.
Determine the moment of inertia, about the generating axis, of the hollow circular tube of mass m obtained by revolving the thin ring shown in the sectional view completely around the generating axis. -R- a
Determine the moments of inertia of the hemispherical shell with respect to the x- and z-axes. The mass of the shell is m, and its thickness is negligible compared with the radius r.
The partial solid of revolution is formed by revolving the shaded area in the x-z plane 90° about the z-axis. If the mass of the solid is m, determine its mass moment of inertia about the z-axis. 22 x = a1- 62 a
A shell of mass m is obtained by revolving the quarter-circular section about the z-axis. If the thickness of the shell is small compared with a and if r = a/3, determine the radius of gyration of the shell about the z-axis. L-
Determine the mass moment of inertia and corresponding radius of gyration of the thin homogeneous parabolic shell about the y-axis. The shell has dimensions r = 70 mm and h = 200 mm, and is made of metal plate having a mass per unit area of 32 kg/m2. h
Every “slender” rod has a finite radius r. Refer to Table D/4 and derive an expression for the percentage error e which results if one neglects the radius r of a homogeneous solid cylindrical rod of length l when calculating its moment of inertia Izz. Evaluate your expression for the ratios r/l
The two small spheres of mass m each are connected by the light rigid rod which lies in the x-z plane. Determine the mass moments of inertia of the assembly about the x-, y-, and z-axes. y 7. m m
The rectangular metal plate has a mass of 15 kg. Compute its moment of inertia about the y-axis. What is the magnitude of the percentage error e introduced by using the approximate relation 1/3ml2 for Ixx? 30 mm 300 mm y 15 mm 100 mm 100 mm
Determine Ixx for the cylinder with a centered circular hole. The mass of the body is m. r2
Determine the mass moment of inertia about the z-axis for the right-circular cylinder with a central longitudinal hole. 2r m 1/7
Determine the moment of inertia of the half-ring of mass m about its diametral axis a-a and about axis b-b through the midpoint of the arc normal to the plane of the ring. The radius of the circular cross section is small compared with r. a m | a
A 6-in. steel cube is cut along its diagonal plane. Calculate the moment of inertia of the resulting prism about the edge x-x. 6" 6" 6"
The uniform coiled spring weighs 4 lb. Approximate its moments of inertia about the x-, y-, and z-axes from the analogy to the properties of a cylindrical shell. y 5" 5"- 3"
Determine the length L of each of the slender rods of mass m/2 which must be centrally attached to the faces of the thin homogeneous disk of mass m in order to make the mass moments of inertia of the unit about the x- and z-axes equal. т L m т 2 L
A badminton racket is constructed of uniform slender rods bent into the shape shown. Neglect the strings and the built-up wooden grip and estimate the mass moment of inertia about the y-axis through O, which is the location of the player’s hand. The mass per unit length of the rod material is ρ.
Calculate the moment of inertia of the steel control wheel, shown in section, about its central axis. There are eight spokes, each of which has a constant cross-sectional area of 200 mm2. What percent n of the total moment of inertia is contributed by the outer rim? 75- 200 mm2- 120 50 100 300 400
The welded assembly is made of a uniform rod which weighs 0.370 lb per foot of length and the semicircular plate which weighs 8 lb per square foot. Determine the mass moments of inertia of the assembly about the three coordinate axes shown. 4" | -6 y
The uniform rod of length 4b and mass m is bent into the shape shown. The diameter of the rod is small compared with its length. Determine the moments of inertia of the rod about the three coordinate axes. y b. b
The welded assembly shown is made from a steel rod which weighs 0.455 lb per foot of length. Determine the mass moment of inertia of the assembly (a) about the y-axis and (b) about the z-axis. 4" -y 6" 6" ---
Calculate the moment of inertia of the solid steel semicylinder about the x-x axis and about the parallel x0-x0 axis. (See Table D/1 for the density of steel.) 100 mm Ox 60 mm 60 mm
The body is constructed of a uniform square plate, a uniform straight rod, a uniform quarter-circular rod, and a particle (negligible dimensions). If each part has the indicated mass, determine the mass moments of inertia of the body about the x-, y-, and z-axes. 9- 0.15m 0.5m b. 0.1m -y 0.25m
The clock pendulum consists of the slender rod of length l and mass m and the bob of mass 7m. Neglect the effects of the radius of the bob and determine IO in terms of the bob position x. Calculate the ratio R of IO evaluated for x = 3/4 l to IO evaluated for x = l. 7m m
Determine the mass moments of inertia of the bracket about the x- and x′-axes. The bracket is made from thin plate of uniform thickness and has a mass of 0.35 kg per square meter of area. y 40 mm / 80 mm 80 mm 100 mm
A square plate with a quarter-circular sector removed has a net mass m. Determine its moment of inertia about axis A-A normal to the plane of the plate. А a y. a a A
The welded assembly consists of a cylindrical shell with a closed semicircular end. The shell is made from sheet metal with a mass of 24 kg/m2, and the end is made from metal plate with a mass of 36 kg/m2. Determine the mass moments of inertia of the assembly about the coordinate axes shown. 600 mm
Determine the mass moment of inertia of the steel bracket about the z-axis which passes through the midline of the base. 2" 3" 1" 1" 1" 1.5" 1.5" 2.
The slender metal rods are welded together in the configuration shown. Each 6-in. segment weighs 0.30 lb. Compute the moment of inertia of the assembly about the y-axis. 6" 6" 6" 6" 6"
The welded assembly is formed from thin sheet metal having a mass of 19 kg/m2. Determine the mass moments of inertia for the assembly about the x- and y-axes. 150 mm 240 mm 80 mm
Determine Ixx for the cone frustum, which has base radii r1 and r2 and mass m. y r2 G
A preliminary design model to ensure rotational stability for a spacecraft consists of the cylindrical shell and the two square panels as shown. The shell and panels have the same thickness and density. It can be shown that rotational stability about the z-axis can be maintained if Izz is less than
Determine the radius of gyration of the aluminum part about the z-axis. The hole in the upper surface is drilled completely through the part. 20. 25 30 5. 10 10- 75 50 -y Dimensions in millimeters
Compute the moment of inertia of the mallet about the O-O axis. The mass of the head is 0.8 kg, and the mass of the handle is 0.5 kg. 60 mm 60 mm 240 mm 60 mm 40 mm dia

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