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engineering
engineering mechanics dynamics

Questions and Answers of Engineering Mechanics Dynamics

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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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.
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
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
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 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
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
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
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
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
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
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 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
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
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
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.
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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.
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
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 =
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
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
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
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
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
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
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
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
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
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,
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
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
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.
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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

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