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engineering
mechanical engineering

Questions and Answers of Mechanical Engineering

A uniform disk of radius 10 r = in. is attached at A to a 26-in. rod AB of negligible weight which can rotate freely in a vertical plane about B. If the rod is displaced 2? from the position shown
A homogeneous rod of weight per unit length equal to 0.3 lb/ft is used to form the assembly shown, which rotates freely about pivot A in a vertical plane. Knowing that the assembly is displaced 2?
A period of 4.1 s is observed for the angular oscillations of a 450-g gyroscope rotor suspended from a wire as shown. Knowing that a period of 6.2 s is obtained when a 50-mm-diameter steel sphere is
A 3-kg slender rod is suspended from a steel wire which is known to have a torsional spring constant 1.95 K = N ?? m/rad. If the rod is rotated through 180? about the vertical and then released,
A 4-lb circular disk of radius r = 40 in. is suspended at its center C from wires AB and BC soldered together at B. The torsional spring constants of the wires are K1 =3 lb ?? ft/rad for AB and K2 =
A 120-lb uniform circular plate is welded to two elastic rods which have fixed ends at supports A and B as shown. The torsional spring constant of each rod is 150 lb ft/?? t/rad, and the system is in
A horizontal platform P is held by several rigid bars which are connected to a vertical wire. The period of oscillation of the platform is found to be 2.2 s when the platform is empty and 3.8 s when
A uniform equilateral triangular plate of side b is suspended from three vertical wires of the same length l. Determine the period of small oscillations of the plate when (a) It is rotated through a
Two blocks, each of mass 1.5 kg, are attached to links which are pin-connected to bar BC as shown. The masses of the links and bar are negligible, and the blocks can slide without friction. Block D
Two blocks, each of mass 1.5 kg, are attached to links which are pin-connected to bar BC as shown. The masses of the links and bar are negligible, and the blocks can slide without friction. Block D
Two small spheres, A and C, each of mass m, are attached to rod AB, which is supported by a pin and bracket at B and by a spring CD of constant k. Knowing that the mass of the rod is negligible and
A 20-lb block is attached to spring A and connected to spring B by a cord and pulley. The block is held in the position shown with both springs un-stretched when the support is removed and the block
The inner rim of a 38-kg flywheel is placed on a knife edge, and the period of its small oscillations is found to be 1.26 s. Determine the centroidal moment of inertia of the flywheel.
A uniform rod AB can rotate in a vertical plane about a horizontal axis at C located at a distance c above the mass center G of the rod. For small oscillations determine the value of c for which the
A connecting rod is supported by a knife edge at point A; the period of its small oscillations is observed to be 0.895 s. The rod is then inverted and supported by a knife edge at point B and the
A thin uniform plate cut into the shape of a quarter circles can rotate in a vertical plane about a horizontal axis at point O. Determine the period of small oscillations of the plate.
A uniform rod ABC weighs 6 lb and is attached to two springs as shown. If end C is given a small displacement and released, determine the frequency of vibration of the rod.
A uniform disk of radius r and mass m can roll without slipping on a cylindrical surface and is attached to bar ABC of length L and negligible mass. The bar is attached to a spring of constant k and
A 7-kg uniform cylinder can roll without sliding on an incline and is attached to a spring AB as shown. If the center of the cylinder is moved 10 mm down the incline and released, determine (a) The
Two uniform rods, each of mass m = 0.6kg and length l = 160 mm, are welded together to form the assembly shown. Knowing that the constant of each spring is k = 120 N/m and that end A is given a small
A slender 10-kg bar AB of length l = 0.6 m is connected to two collars of negligible weight. Collar A is attached to a spring of constant k = 1.5kN/m and can slide on a horizontal rod, while collar B
A slender 5-kg bar AB of length l = 0.6 m is connected to two collars, each of mass 2.5 kg. Collar A is attached to a spring of constant k = 1.5kN/m and can slide on a horizontal rod, while collar B
Three identical 3.6-kg uniform slender bars are connected by pins as shown and can move in a vertical plane. Knowing that bar BC is given a small displacement and released, determine the period of
A 0.7-kg sphere A and a 0.5-kg sphere C are attached to the ends of a 1-kg rod AC which can rotate in a vertical plane about an axis at B. Determine the period of small oscillations of the rod.
Spheres A and C, each of weight W, are attached to the ends of a homogeneous rod of the same weight W and of length 2l which is bent as shown. The system is allowed to oscillate about a frictionless
The 3-lb rod AB is bolted to a 5-lb disk. Knowing that the disk rolls without sliding, determine the period of small oscillations of the system.
Two 6-kg uniform disks are attached to the 9-kg rod AB as shown. Knowing that the constant of the spring is 5kN/m and that the disks roll without sliding, determine the frequency of vibration of the
A half section of a uniform cylinder of radius r and mass m rests on two casters A and B, each of which is a uniform cylinder of radius r/4 and mass m/8. Knowing that the half cylinder is rotated
The 10-kg rod AB is attached to two 4-kg disks as shown. Knowing that the disks roll without sliding, determine the frequency of small oscillations of the system.
Three collars, each of mass m, are connected by pins to bars AC and BC, each of length l and negligible mass. Collars A and B can slide without friction on a horizontal rod and are connected by a
Two 6-lb uniform semicircular plates are attached to the 4-lb rod AB as shown. Knowing that the plates roll without sliding, determine the period of small oscillations of the system.
A uniform 6-lb disk can roll without slipping on a cylindrical surface and is attached to a 4-lb uniform slender bar AB. The bar is attached to a spring of constant k = 20 lb/ft and can rotate freely
A slender rod of mass m and length l is suspended from two vertical springs, each of constant k as shown. The rod is in equilibrium when it is given a small rotation about a horizontal axis through G
A section of uniform pipe is suspended from two vertical cables attached at A and B. Determine the frequency of oscillation when the pipe is given a small rotation about the centroidal axis OO?? and
A half section of pipe is placed on a horizontal surface, rotated through a small angle, and then released. Assuming that the pipe section rolls without sliding, determine the period of oscillation
A thin plate of length l rests on a half cylinder of radius r. Derive an expression for the period of small oscillations of the plate.
(a) Neglecting fluid friction, determine the period of vibration of the shell when it is displaced vertically and then released. (b) Solve part a, assuming that the tank is accelerated upward at the
A 4-kg collar can slide on a frictionless horizontal rod and is attached to a spring of constant 450 N/m. It is acted upon by a periodic force of magnitude P = Pm sin ωft where Pm = 13N. Determine
A 4-kg collar can slide on a frictionless horizontal rod and is attached to a spring of constant k. It is acted upon by a periodic force of magnitude P = Pm sin, ωft where 9 Pm = N and 5 ωf =
A collar of mass m which slides on a frictionless horizontal rod is attached to a spring of constant k and is acted upon by a periodic force of magnitude P = Pm sin ωft. Determine the range of
A small 40-lb block A is attached to the rod BC of negligible mass which is supported at B by a pin and bracket and at C by a spring of constant k = 140 lb/ft. The system can move in a vertical plane
A small 40-lb block A is attached to the rod BC of negligible mass which is supported at B by a pin and bracket and at C by a spring of constant k = 140 lb/ft. The system can move in a vertical plane
The 1.2-kg bob of a simple pendulum of length l = 600 mm is suspended from a 1.4-kg collar C. Knowing that the collar is acted upon by a periodic force P = Pm sin ωft, where Pm = 0.5N and ωf =
A cantilever beam AB supports a block which causes a static deflection of 40 mm at B. Assuming that the support at A undergoes a vertical periodic displacement δ = δm sin ωft, where δm = 10mm,
A 2-kg block A slides in a vertical frictionless slot and is connected to a moving support B by means of a spring AB of constant k = 117N/m. Knowing that the displacement of the support is δ = δm
A 16-lb block A slides in a vertical frictionless slot and is connected to a moving support B by means of a spring AB of constant 130 lb/k = ft. Knowing that the displacement of the support is δ =
A 5-lb block A is attached to a spring of constant k = 4 lb/ft and a bar BCD of negligible weight. The bar is connected at D to a moving support E by means of an identical spring. Knowing that the
A small 2-kg sphere B is attached to the bar AB of negligible mass which is supported at A by a pin and bracket and connected at C to a moving support D by means of a spring of constant k = 3.6kN/m.
A beam ABC is supported by a pin connection at A and by rollers at B. A 120-kg block placed on the end of the beam causes a static deflection of 15 mm at C. Assuming that the support at A undergoes a
A simple pendulum of length l is suspended from a collar C which is forced to move horizontally according to the relation xc = Î´m sin Ï‰ft. Determine the range of values of
In Prob. 19.110, determine the range of values of ωf for which the amplitude of the motion of the bob exceeds 2δm.
A 200-kg motor is supported by springs having a total constant of 215kN/m. The unbalance of the rotor is equivalent to a 30-g mass located 200 mm from the axis of rotation. Determine the range of
A motor of mass 18 kg is supported by four springs, each of constant 40kN/m. The motor is constrained to move vertically, and the amplitude of its motion is observed to be 1.5 mm at a speed of 1200
A 360-lb motor is bolted to a light horizontal beam. The unbalance of its rotor is equivalent to a 0.9-oz weight located 7.5 in. from the axis of rotation, and the static deflection of the beam due
A motor of mass M is supported by springs with an equivalent spring constant k. The unbalance of its rotor is equivalent to a mass m located at a distance r from the axis of rotation. Show that when
Rod AB is rigidly attached to the frame of a motor running at a constant speed. When a collar of mass m is placed on the spring, it is observed to vibrate with an amplitude of 15 mm. When two
Solve Prob. 19.116, assuming that the speed of the motor is changed and that one collar has an amplitude of 9 mm and two collars have an amplitude of 3 mm.
The unbalance of the rotor of a 400-lb motor is equivalent to a 0.8-oz weight located 6 in. from the axis of rotation. Knowing that the motor is supported by four springs, each of constant 5 kips/ft,
A counter-rotating eccentric mass exciter consisting of two rotating 3.5-oz weights describing circles of radius r at the same speed but in opposite senses is placed on a machine element to induce a
Figures (1) and (2) show how springs can be used to support a block in two different situations, in Fig. (1) They help decrease the amplitude of the fluctuating force transmitted by the block to the
A 27-kg disk is attached with an eccentricity e = 150μm to the midpoint of a vertical shaft AB which revolves at a constant angular velocity ωf. Knowing that the spring constant k for horizontal
A small trailer and its load have a total weight of 500 lb. The trailer is supported by two springs, each of constant 350 lb/ft, and is pulled over a road, the surface of which can be approximated by
Block A can move without friction in the slot as shown and is acted upon by a vertical periodic force of magnitude P = Pm sin ωft where ωf = 2 rad/s and Pm = 5lb. A spring of constant k is attached
A vibrometer used to measure the amplitude of vibrations consists of a box containing a mass-spring system with a known natural frequency of 150 Hz. The box is rigidly attached to a surface which is
A certain accelerometer consists essentially of a box containing a mass-spring system with a known natural frequency of 1760 Hz. The box is rigidly attached to a surface which is moving according to
Show that in the case of heavy damping (c > cc), a body never passes through its position of equilibrium O(a) If it is released with no initial velocity from an arbitrary position, or(b) If it is
Show that in the case of heavy damping (c > cc), a body released from an arbitrary position with an arbitrary initial velocity cannot pass more than once through its equilibrium position.
In the case of light damping, the displacements x1, x2, x3, shown in Fig. 19.11, may be assumed equal to the maximum displacements. Show that the ratio of any two successive maximum displacements xn
In practice, it is often difficult to determine the logarithmic decrement of a system with light damping defined in Prob. 19.128 by measuring two successive maximum displacements. Show that the
In a system with light damping (c < cc), the period of vibration is commonly defined as the time interval τd = 2π/ωd corresponding to two successive points where the displacement-time curve
Successive maximum displacements of a spring-mass-dashpot system similar to that shown in Fig. 19.10 are 1.25, 0.75, and 0.45 in. Knowing that W = 36 lb and k = 175 lb/ft, determine(a) The damping
A 2-kg block is supported by a spring of constant k = 128 N/m and a dashpot with a coefficient of viscous damping c = 0.6 N ?? s/m. The block is in equilibrium when it is struck from below by a
The barrel of a field gun weighs 1800 lb and is returned into firing position after recoil by a recuperator of constant c = 1320 lb ⋅ s/ft. Determine(a) The constant k which should be used for the
A 0.9-kg block B is connected by a cord to a 2.4-kg block A which is suspended as shown from two springs, each of constant 180 k = N/m, and a dashpot of damping coefficient c = 7.5N ?? s/m. Knowing
A 0.9-kg block B is connected by a cord to a 2.4-kg block A which is suspended as shown from two springs, each of constant k = 180 N/m, and a dashpot of damping coefficient c = 60N ?? s/m. Knowing
A 1.8-kg uniform rod is supported by a pin at O and a spring at A and is connected to a dashpot at B. Determine (a) The differential equation of motion for small oscillations, (b) The angle that the
A 1.8-kg uniform rod is supported by a pin at O and a spring at A and is connected to a dashpot at B. Determine (a) The differential equation of motion for small oscillations, (b) The angle that the
A platform of weight 200lb, supported by two springs each of constant k = 250 lb/in., is subjected to a periodic force of maximum magnitude equal to 125lb. Knowing that the coefficient of damping is
Solve Prob. 19.138, assuming that the coefficient of damping is increased to 12lb ⋅ s/in.
In the case of the forced vibration of a system, determine the range of values of the damping factor c/cc for which the magnification factor will always decrease as the frequency ratio ωf/ωn
Show that for a small value of the damping factor c/cc, the maximum amplitude of a forced vibration occurs when ωf ≈ ωn and that the corresponding value of the magnification factor is ½ (cc/c)
A 36-lb motor is bolted to a light horizontal beam which has a static deflection of 0.075 in. due to the weight of the motor. Knowing that the unbalance of the rotor is equivalent to a weight of 0.64
A 45-kg motor is bolted to a light horizontal beam which has a static deflection of 6 mm due to the weight of the motor. The unbalance of the motor is equivalent to a mass of 110 g located 75 mm from
Solve Prob. 19.113, assuming that a dashpot having a coefficient of damping c = 350N ?? s/m has been connected to the motor and to the ground.
The unbalance of the rotor of a 180-kg motor is equivalent to a mass of 85 g located 150 mm from the axis of rotation. The pad which is placed between the motor and the foundation is equivalent to a
A 200-lb motor is supported by two springs, each of constant 15 kips/ft, and is connected to the ground by a dashpot having a coefficient of damping c = 490lb ?? s/ft. The motor is constrained to
A machine element is supported by springs and is connected to a dashpot as shown. Show that if a periodic force of magnitude P = Pm sin ωft is applied to the element, the amplitude of the
A 91-kg machine element supported by four springs, each of constant 175 N/m, is subjected to a periodic force of frequency 0.8 Hz and amplitude 89 N. Determine the amplitude of the fluctuating force
For a steady-state vibration with damping under a harmonic force, show that the mechanical energy dissipated per cycle by the dashpot is E = πcx2mωf, where c is the coefficient of damping, xm is
The suspension of an automobile can be approximated by the simplified spring-dashpot system shown.(a) Write the differential equation defining the vertical displacement of the mass m when the system
Two blocks A and B, each of mass m, are supported as shown by three springs of the same constant k. Blocks A and B are connected by a dashpot and block B is connected to the ground by two dashpots,
Express in terms of L, C, and E the range of values of the resistance R for which oscillations will take place in the circuit shown when switch S isclosed.
Consider the circuit of Prob. 19.152 when the capacitor C is removed. If switch S is closed at time t = 0, determine(a) The final value of the current in the circuit,(b) The time t at which the
Draw the electrical analogue of the mechanical system shown.
Draw the electrical analogue of the mechanical system shown.
Write the differential equations defining (a) The displacements of the mass m and of the point A, (b) The charges on the capacitors of the electrical analogue.
Write the differential equations defining (a) The displacements of the mass m and of the point A, (b) The charges on the capacitors of the electrical analogue.
The bob of a simple pendulum of length l = 40 in. is released from rest when θ = +5?. Assuming simple harmonic motion, determine 1.6 s after release (a) The angle θ, (b) The magnitudes of the
A 50-kg block is supported by the spring arrangement shown. The block is moved vertically downward from its equilibrium position and released. Knowing that the amplitude of the resulting motion is 60
A rod of mass m and length L rests on two pulleys A and B which rotate in opposite directions as shown. Denoting by μk the coefficient of kinetic friction between the rod and the pulleys, determine

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