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engineering
engineering mechanics dynamics

Questions and Answers of Engineering Mechanics Dynamics

Determine the products of inertia about the coordinate axes for the unit which consists of four small particles, each of mass m, connected by the light but rigid slender rods. 9- 9- m т x m
Determine the products of inertia about the coordinate axes for the unit which consists of three small spheres, each of mass m, connected by the light but rigid slender rods. m m | y m
Determine the product of inertia Ixy for the slender rod of mass m. y 1/2 1/2
Determine the products of inertia of the uniform slender rod of mass m about the coordinate axes shown. т h b
Determine the products of inertia about the coordinate axes for the thin square plate with two circular holes. The mass of the plate material per unit area is ρ. y 이4.6_4|614.6-4 |
Determine the products of inertia about the coordinate axes for the thin plate of mass m which has the shape of a circular sector of radius a and angle β as shown. y | m a
The homogeneous plate of Prob. B/7 is repeated here. Determine the product of inertia for the plate about the x-y axes. The plate has mass m and uniform thickness t. y x = ky? h Thickness t
Determine by direct integration the product of inertia of the thin homogeneous triangular plate of mass m about the x-y axes. Then, use the parallel-axis theorem to determine the product of inertia
The aluminium casting consists of a 6-in. cube with a 4-in. cubical recess. Calculate the products of inertia of the casting about the axes shown. 4" 6" 4" 6" 6
The S-shaped piece is formed from a rod of diameter d and bent into the two semicircular shapes. Determine the products of inertia for the rod, for which d is small compared with r. d トく
Determine the products of inertia about the coordinate axes for the assembly which consists of uniform slender rods. Each rod has a mass ρ per unit length. a a a a
Prove that the moment of inertia of the rigid assembly of three identical balls, each of mass m and radius r, has the same value for all axes through O. Neglect the mass of the connecting rods. m m -
The plane of the thin circular disk of mass m and radius r makes an angle β with the x-z plane. Determine the product of inertia of the disk with respect to the y-z plane. yー 1. G
The L-shaped piece is cut from steel plate having a mass per unit area of 160 kg/m2. Determine and plot the moment of inertia of the piece about axis A-A as a function of θ from θ = 0 to θ = 90°
Determine the moment of inertia I about axis O-M for the uniform slender rod bent into the shape shown. Plot I versus θ from θ = 0 to θ = 90° and determine the minimum value of I and the angle a
Determine the inertia tensor for the homogeneous thin plate about the x-, y-, and z-axes. The plate has a mass m and uniform thickness t. What is the minimum angle, measured from the x-axis, which
The assembly of three small spheres connected by light rigid bars of Prob. B/58 is repeated here. Determine the principal moments of inertia and the direction cosines associated with the axis of
The thin plate has a mass ρ per unit area and is formed into the shape shown. Determine the principal moments of inertia of the plate about axes through O. 26 -y b
The slender rod has a mass ρ per unit length and is formed into the shape shown. Determine the principal moments of inertia about axes through O and calculate the direction cosines of the axis of
The welded assembly is formed from uniform sheet metal with a mass of 32 kg/m2. Determine the principal mass moments of inertia for the assembly and the corresponding direction cosines for each
Derive the equation of motion for the system shown in terms of the displacement x. The masses are coupled through the light connecting rod ABC which pivots about the smooth bearing at point O.
The mass of the system shown is released from rest at x0 = 6 in. when t = 0. Determine the displacement x at t = 0.5 sec if (a) c = 12 lb-sec/ft (b) c = 18 lb-sec/ft. W = 96.6 lb www. k = 1
The mass of a g en critically damped system is released at time t-o rrom the posit on xo = 40 mm with a negative initial velocity. The mass m is 13 kg and the spring constant k is 836 N m. Determine
Determine the equation of motion for the system in terms of the variable x. The cables remain taut at all times, and the pulleys turn independently. Neglect friction and the mass of the pulleys.
The system shown is released from rest from an initial position x0. Determine the overshoot displacement x1. Assume translational motion in the x-direction. k = 108 N/m c 18 N.s/m m = 3 kg t X1
Derive the differential equation of motion for the system shown in terms of the variable x1. Neglect friction and the mass of the linkage. >x1 k1 C1 А www X2 k2 m2 C2 B
Further design refinement for the weighing platform of Prob. 8/15 is shown here where two viscous dampers are to be added to limit the ratio of successive positive amplitudes of vertical vibration in
Determine the damping ratio ζ for the system shown. The system parameters are m = 4 kg, k = 500 N/m, and c = 100 N∙ s/m. Neglect the mass and friction of all pulleys, and assume that the cord
A linear harmonic oscillator having a mass of 1.10 kg is set into motion with viscous damping. If the frequency is 10 Hz and if two successive amplitudes a full cycle apart are measured to be 4.65 mm
The figure represents the measured displacement-time relationship for a vibration with small damping where it is impractical to achieve accurate results by measuring the nearly equal amplitudes of
The period τd of damped linear oscillation for a certain 1-kg mass is 0.3 s. If the stiffness of the supporting linear spring is 800 N/m, calculate the damping coefficient c.
Determine the value of the viscous damping coefficient c for which the system shown is critically damped. 30 kN/m 35 kg ww
The addition of damping to an undamped spring-mass system causes its period to increase by 25 percent. Determine the damping ratio ζ.
Viscous damping is added to an initially undamped spring-mass system. For what value of the damping ratio ζ will the damped natural frequency wd be equal to 90 percent of the natural frequency of
Determine the value of the damping ratio ζ for the simple spring-mass-dashpot system shown. c = 2.5 lb-sec/ft 8 lb www k = 3 lb/in.
The slider of mass m is confined to the horizontal slot shown. The two springs each of constant k are linear. Derive the nonlinear equation of motion for small values of y, retaining terms of order
Calculate the natural circular frequency wn of the system shown in the figure. The mass and friction of the pulleys are negligible. k m 2m
Shown in the figure is a model of a one-story building. The bar of mass m is supported by two light elastic upright columns whose upper and lower ends are fixed against rotation. For each column, if
A 90-kg man stands at the end of a diving board and causes a vertical oscillation which is observed to have a period of 0.6 s. What is the static deflection δst at the end of the board? Neglect the
An energy-absorbing car bumper with its springs initially undeformed has an equivalent spring constant of 3000 lb/in. If the 2500-lb car approaches a massive wall with a speed of 5 mi/hr,
If both springs are unstretched when the mass is in the central position shown, determine the static deflection δst of the mass. What is the period of oscillatory motion about the position of static
With the assumption of no slipping, determine the mass m of the block which must be placed on the top of the 6-kg cart in order that the system period be 0.75 s. What is the minimum coefficient μs
Replace the springs in each of the two cases shown by a single spring of stiffness k (equivalent spring stiffness) which will cause each mass to vibrate with its original frequency. k2 k2 k1 k1 (b)
Calculate the natural frequency ƒn of vibration if the mass is deflected from its equilibrium position and released from rest. Each pair of springs is connected by an inextensible cable. Evaluate
During the design of the spring-support system for the 4000-kg weighing platform, it is decided that the frequency of free vertical vibration in the unloaded condition shall not exceed 3 cycles per
The 4-oz slider oscillates in the fixed slot under the action of the three springs, each of stiffness k = 0.5 lb/in. If the initial conditions at time t = 0 are x0 = 0.1 in. and x˙0 = 0.5 in./sec,
An old car being moved by a magnetic crane pickup is dropped from a short distance above the ground. Neglect any damping effects of its worn-out shock absorbers and calculate the natural frequency
Determine the natural frequency in radians per second for the system shown. Neglect the mass and friction of the pulleys. m
In the equilibrium position, the 30-kg cylinder causes a static deflection of 50 mm in the coiled spring. If the cylinder is depressed an additional 25 mm and released from rest, calculate the
Determine the period τ for the system shown. The cable is always taut, and the mass and friction of the pulley are to be neglected. m
The vertical plunger has a mass of 2.5 kg and is supported by the two springs, which are always in compression. Calculate the natural frequency ƒn of vibration of the plunger if it is deflected from
Determine the natural frequency in cycles per second for the system shown. Neglect the mass and friction of the pulleys. Assume that the block of mass m remains horizontal. m
For the system of Prob. 8/2, determine the position x as a function of time if the mass is released at time t = 0 from a position 2 inches to the right of the equilibrium position with an initial
Determine the natural frequency of the spring-mass system in both radians per second and cycles per second (Hz). k = 54 lb/in. www 64.4 lb
When a 3-kg collar is placed upon the pan which is attached to the spring of unknown constant, the additional static deflection of the pan is observed to be 42 mm. Determine the spring constant k in
The half-cylindrical shell of mass m, radius r, and length b revolves about one edge along the z-axis with a constant rate w as shown. Determine the angular momentum H of the shell with respect to
Each of the quarter-circular plates has a mass of 2 kg and is secured to the vertical shaft mounted in the fixed bearing at O. Calculate the magnitude M of the bending moment in the shaft at O for a
Each of the two right-angle bent rods weighs 2.80 lb and is parallel to the horizontal x-y plane. The rods are welded to the vertical shaft, which rotates about the z-axis with a constant angular
The dynamic imbalance of a certain crankshaft is approximated by the physical model shown, where the shaft carries three small 1.5-lb spheres attached by rods of negligible mass. If the shaft rotates
The uniform circular disk of 4-in. radius and small thickness weighs 8 lb and is spinning about its y′- axis at the rate N = 300 rev/min with its plane of rotation tilted at a constant angle β =
A top consists of a ring of mass m = 0.52 kg and mean radius r = 60 mm mounted on its central pointed shaft with spokes of negligible mass. The top is given a spin velocity of 10 000 rev/min and
The thin circular disk of mass m and radius R is hinged about its horizontal tangent axis to the end of a shaft rotating about its vertical axis with an angular velocity w. Determine the steady-state
The uniform square plate of mass m is welded at O to the end of the shaft, which rotates about the vertical z-axis with a constant angular velocity w. Determine the moment applied to the plate by the
The uniform slender rod of length l is welded to the bracket at A on the underside of the disk B. The disk rotates about a vertical axis with a constant angular velocity w. Determine the value of w
The half-cylindrical shell of radius r, length 2b, and mass m revolves about the vertical z-axis with a constant angular velocity w as indicated. Determine the magnitude M of the bending moment in
The homogeneous thin triangular plate of mass m is welded to the horizontal shaft, which rotates freely in the bearings at A and B. If the plate is released from rest in the horizontal position
A dynamics instructor demonstrates gyroscopic principles to his students. He suspends a rapidly spinning wheel with a string attached to one end of its horizontal axle. Describe the precession motion
The student has volunteered to assist in a classroom demonstration involving a momentum wheel which is rapidly spinning with angular speed p as shown. The instructor has asked her to hold the axle of
A car makes a turn to the right on a level road. Determine whether the normal reaction under the right rear wheel is increased or decreased as a result of the gyroscopic effect of the precessing
The 50-kg wheel is a solid circular disk which rolls on the horizontal plane in a circle of 600-mm radius. The wheel shaft is pivoted about the axis O-O and is driven by the vertical shaft at the
The special-purpose fan is mounted as shown. The motor armature, shaft, and blades have a combined mass of 2.2 kg with radius of gyration of 60 mm. The axial position b of the 0.8-kg block A can be
An airplane has just cleared the runway with a takeoff speed v. Each of its freely spinning wheels has a mass m, with a radius of gyration k about its axle. As seen from the front of the airplane,
An experimental antipollution bus is powered by the kinetic energy stored in a large flywheel which spins at a high speed p in the direction indicated. As the bus encounters a short upward ramp, the
The 210-kg rotor of a turbojet aircraft engine has a radius of gyration of 220 mm and rotates counterclockwise at 18 000 rev/min as viewed from the front. If the aircraft is traveling at 1200 km/h
A small air compressor for an aircraft cabin consists of the 3.50-kg turbine A which drives the 2.40-kg blower B at a speed of 20 000 rev/min. The shaft of the assembly is mounted transversely to the
The two solid cones with the same base and equal altitudes are spinning in space about their common axis at the rate p. For what ratio h/r will precession of their spin axis be impossible? h キャー
The blades and hub of the helicopter rotor weigh 140 lb and have a radius of gyration of 10 ft about the z-axis of rotation. With the rotor turning at 500 rev/min during a short interval following
The 4-oz top with radius of gyration about its spin axis of 0.62 in. is spinning at the rate p = 3600 rev/min in the sense shown, with its spin axis making an angle θ = 20° with the vertical. The
The figure shows a gyro mounted with a vertical axis and used to stabilize a hospital ship against rolling. The motor A turns the pinion which precesses the gyro by rotating the large precession gear
Each of the identical wheels has a mass of 4 kg and a radius of gyration kz = 120 mm and is mounted on a horizontal shaft AB secured to the vertical shaft at O. In case (a) The horizontal shaft
The figure shows the side view of the wheel carriage (truck) of a railway passenger car where the vertical load is transmitted to the frame in which the journal wheel bearings are located. The lower
The primary structure of a proposed space station consists of five spherical shells connected by tubular spokes. The moment of inertia of the structure about its geometric axis A-A is twice as much
The uniform 640-mm rod has a mass of 3 kg and is welded centrally to the uniform 160-mm-radius circular disk which has a mass of 8 kg. The unit is given a spin velocity p = 60 rad/s in the direction
The electric motor has a total weight of 20 lb and is supported by the mounting brackets A and B attached to the rotating disk. The armature of the motor has a weight of 5 lb and a radius of gyration
The spacecraft shown is symmetrical about its z-axis and has a radius of gyration of 720 mm about this axis. The radii of gyration about the x- and y-axes through the mass center are both equal to
The 8-lb rotor with radius of gyration of 3 in. rotates on ball bearings at a speed of 3000 rev/min about its shaft OG. The shaft is free to pivot about the X-axis, as well as to rotate about the
The housing of the electric motor is freely pivoted about the horizontal x-axis, which passes through the mass center G of the rotor. If the motor is turning at the constant rate Φ˙ = p, determine
The thin ring is projected into the air with a spin velocity of 300 rev/min. If its geometric axis is observed to have a very slight precessional wobble, determine the frequency ƒ of the wobble. 300
A boy throws a thin circular disk (like a Frisbee) with a spin rate of 300 rev/min. The plane of the disk is seen to wobble through a total angle of 10°. Calculate the period τ of the wobble and
The figure shows a football in three common inflight configurations. Case (a) is a perfectly thrown spiral pass with a spin rate of 120 rev/min. Case (b) is a wobbly spiral pass again with
The rectangular bar is spinning in space about its longitudinal axis at the rate p = 200 rev/min. If its axis wobbles through a total angle of 20° as shown, calculate the period τ of the
The 5-kg disk and hub A have a radius of gyration of 85 mm about the z0-axis and spin at the rate p = 1250 rev/min. Simultaneously, the assembly rotates about the vertical z-axis at the rate Ω = 400
The uniform slender bar of mass m and length l is centrally mounted on the shaft A-A, about which it rotates with a constant speed Φ˙ = p. Simultaneously, the yoke is forced to rotate about the
The solid circular disk of mass m and small thickness is spinning freely on its shaft at the rate p. If the assembly is released in the vertical position at θ = 0 with θ˙ = 0, determine the
The two solid homogeneous right-circular cones, each of mass m, are fastened together at their vertices to form a rigid unit and are spinning about their axis of radial symmetry at the rate p = 200
The solid cylindrical rotor weighs 64.4 lb and is mounted in bearings A and B of the frame which rotates about the vertical Z-axis. If the rotor spins at the constant rate p = 50 rad/sec relative to
The cylindrical shell is rotating in space about its geometric axis. If the axis has a slight wobble, for what ratios of l/r will the motion be direct or retrograde precession?
The solid cube of mass m and side a revolves about an axis M-M through a diagonal with an angular velocity w. Write the expression for the angular momentum H of the cube with respect to the axes
An experimental car is equipped with a gyro stabilizer to counteract completely the tendency of the car to tip when rounding a curve (no change in normal force between tires and road). The rotor of
The wheels of the jet plane are spinning at their angular rate corresponding to a takeoff speed of 150 km/h. The retracting mechanism operates with θ increasing at the rate of 30° per second.

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