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engineering
engineering mechanics dynamics

Questions and Answers of Engineering Mechanics Dynamics

A finite volume scheme is developed using a two-dimensional structured Cartesian grid with constant steps \(\Delta x\) and \(\Delta y\) (see Figure 5.5a). Write the following approximations using the
The Burgers equation \(u_{t}+u u_{x}=\mu u_{x x}\) is solved by the finite volume method on a one-dimensional grid with constant step \(\Delta x\) (see Figure 5.5b). Develop the schemes based on the
The conservation equation (5.1) is solved by the finite volume method on a two-dimensional structured Cartesian grid with constant steps \(\Delta x\) and \(\Delta y\). Develop the schemes based on
The two-dimensional heat equation \(\partial u / \partial t=abla \cdot(\chi abla u)+f\), where \(\chi\) is a function of \(u\) and \(f\) is a known function of \(\boldsymbol{x}\) and \(t\), is solved
Formulate complete PDE problems (specify the equation, space domain, time interval, and boundary and initial conditions) for the following model situations.a) Conduction heat transfer occurs in a
What are the characteristics of a quasi-linear equation of second order? How are they determined?
Classify (determine the type of) the following PDEs of second order: = 0, (x2 1) +2024 = 25x3 - 1) 25(x-1) + + = cos(5t).
Determine the type (hyperbolic, parabolic, or elliptic) of each PDE in Problem 1.'Problem 1Formulate complete PDE problems (specify the equation, space domain, time interval, and boundary and initial
Transform the one-dimensional wave equation (3.4) into a system of PDE of the first order, such as (3.26). Find the eigenvalues of matrix \(\boldsymbol{R}\), and determine the type of the system.
Conduct the same analysis as in the previous problem for the Poisson equation (3.8). + = f(x, y). (3.8)
Consider the equationwhere \(u=u(x, t)\) and \(c\) and \(\mu\) are positive constants. Would you expect hyperbolic (wavelike) or parabolic (diffusion-like) behavior of the solution? Ot + c = - (3.45)
Define the domain of dependence and domain of influence of a point \(\mathrm{P}\) in a solution of a PDE? How are these domains determined for each type of the PDE of second order?
Verify coupling of the linear algebraic equations of the system (3.42)-(3.44) by writing the discretization formula for the interior grid points \(\left(x_{i}, t^{n}\right),\left(x_{i+1},
Write the formula for the material derivative of the specific internal energy \(e\) of a flowing fluid. What does it represent?
What is Newtonian fluid?
What are the models of incompressible fluid and ideal gas?
Verify that the Navier-Stokes equations (2.19) reduce to (2.22) in the case of a flow with constant density and viscosity. [(+)]+[(+3)]+ V.V+2- - or + 3/2 [ " ( - + 2 du)] Pfz ze ze e
Write the integral equation for a fixed control volume \(\Omega\) that expresses the principle of conservation of chemical species described in Section 2.2.2. Let us now assume that the fluid is a
Following the procedure described in Section 2.5 derive the continuity equation (2.6) from the integral mass conservation equation (2.34). t +V Vp+pV.V= +V (V) = 0. (2.6) at
Consider the following problems of conduction heat transfer within a solid body. For each problem, write the entire system of governing equations and boundary conditions. Assume constant physical
Define the computational domain, and write the full system of governing equations and boundary conditions for the following situations. In each of them, consider a long straight duct with smooth
The belt-driven pulley and attached disk are rotating with increasing angular velocity. At a certain instant the speed \(v\) of the belt is \(1.5 \mathrm{~m} / \mathrm{s}\), and the total
A clockwise variable torque is applied to a flywheel at time \(t=0\) causing its clockwise angular acceleration to decrease linearly with angular displacement \(\theta\) during 20 revolutions of the
Point \(A\) of the circular disk is at the angular position \(\theta=0\) at time \(t=0\). The disk has angular velocity \(\omega_{0}=\) \(0.1 \mathrm{rad} / \mathrm{s}\) at \(t=0\) and subsequently
The telescoping link is hinged at \(O\), and its end \(A\) is given a constant upward velocity of \(200 \mathrm{~mm} / \mathrm{s}\) by the piston rod of the fixed hydraulic cylinder \(B\). Calculate
For the instant represented when \(y=160 \mathrm{~mm}\), the piston rod of the hydraulic cylinder \(C\) imparts a vertical motion to the pin \(B\) consisting of \(\dot{y}=400 \mathrm{~mm} /
The rod \(A B\) slides through the pivoted collar as end \(A\) moves along the slot. If \(A\) starts from rest at \(x=0\) and moves to the right with a constant acceleration of \(4 \mathrm{in}\)./
Motion of the bar is controlled by the constrained paths of \(A\) and \(B\). If the angular velocity of the bar is \(2 \mathrm{rad} / \mathrm{s}\) counterclockwise as the position
The hydraulic cylinder produces a limited horizontal motion of point \(A\). If \(v_{A}=4 \mathrm{~m} / \mathrm{s}\) when \(\theta=45^{\circ}\), determine the magnitude of the velocity of \(D\) and
The center \(O\) of the wheel is mounted on the sliding block, which has an acceleration \(a_{O}=8 \mathrm{~m} / \mathrm{s}^{2}\) to the right. At the instant when \(\theta=45^{\circ}, \dot{\theta}=3
The center \(O\) of the disk has the velocity and acceleration shown in the figure. If the disk rolls without slipping on the horizontal surface, determine the velocity of \(A\) and the acceleration
The pickup truck weighs \(3220 \mathrm{lb}\) and reaches a speed of \(30 \mathrm{mi} / \mathrm{hr}\) from rest in a distance of \(200 \mathrm{ft}\) up the 10-percent incline with constant
The vertical bar \(A B\) has a mass of \(150 \mathrm{~kg}\) with center of mass \(G\) midway between the ends. The bar is elevated from rest at \(\theta=0\) by means of the parallel links of
The concrete block weighing \(644 \mathrm{lb}\) is elevated by the hoisting mechanism shown, where the cables are securely wrapped around the respective drums. The drums, which are fastened together
The pendulum has a mass of \(7.5 \mathrm{~kg}\) with center of mass at \(G\) and has a radius of gyration about the pivot \(O\) of \(295 \mathrm{~mm}\). If the pendulum is released from rest at
A metal hoop with a radius \(r=6\) in. is released from rest on the \(20^{\circ}\) incline. If the coefficients of static and kinetic friction are \(\mu_{s}=0.15\) and \(\mu_{k}=0.12\), determine the
The \(\operatorname{drum} A\) is given a constant angular acceleration \(\alpha_{0}\) of \(3 \mathrm{rad} / \mathrm{s}^{2}\) and causes the 70-kg spool \(B\) to roll on the horizontal surface by
The slender bar \(A B\) weighs \(60 \mathrm{lb}\) and moves in the vertical plane, with its ends constrained to follow the smooth horizontal and vertical guides. If the 30-lb force is applied at
A car door is inadvertently left slightly open when the brakes are applied to give the car a constant rearward acceleration \(a\). Derive expressions for the angular velocity of the door as it swings
The wheel rolls up the incline on its hubs without slipping and is pulled by the \(100-\mathrm{N}\) force applied to the cord wrapped around its outer rim. If the wheel starts from rest, compute its
The 4 -ft slender bar weighs \(40 \mathrm{lb}\) with mass center at \(B\) and is released from rest in the position for which \(\theta\) is essentially zero. Point \(B\) is confined to move in the
In the mechanism shown, each of the two wheels has a mass of \(30 \mathrm{~kg}\) and a centroidal radius of gyration of \(100 \mathrm{~mm}\). Each link \(O B\) has a mass of \(10 \mathrm{~kg}\) and
The movable rack \(A\) has a mass of \(3 \mathrm{~kg}\), and rack \(B\) is fixed. The gear has a mass of \(2 \mathrm{~kg}\) and a radius of gyration of \(60 \mathrm{~mm}\). In the position shown, the
A constant force \(P\) is applied to end \(A\) of the two identical and uniform links and causes them to move to the right in their vertical plane with a horizontal acceleration \(a\). Determine the
The force \(P\), which is applied to the cable wrapped around the central hub of the symmetrical wheel, is increased slowly according to \(P=1.5 t\), where \(P\) is in pounds and \(t\) is the time in
The sheave \(E\) of the hoisting rig shown has a mass of \(30 \mathrm{~kg}\) and a centroidal radius of gyration of \(250 \mathrm{~mm}\). The \(40-\mathrm{kg}\) load \(D\) which is carried by the
The uniform rectangular block of dimensions shown is sliding to the left on the horizontal surface with a velocity v when it strikes the small step at O. Assume negligible rebound at the step and
The 0.8-m arm \(O A\) for a remote-control mechanism is pivoted about the horizontal \(x\)-axis of the clevis, and the entire assembly rotates about the \(z\)-axis with a constant speed \(N=60
The electric motor with an attached disk is running at a constant low speed of \(120 \mathrm{rev} / \mathrm{min}\) in the direction shown. Its housing and mounting base are initially at rest. The
Crank \(C B\) rotates about the horizontal axis with an angular velocity \(\omega_{1}=6 \mathrm{rad} / \mathrm{s}\) which is constant for a short interval of motion which includes the position shown.
Determine the angular acceleration \(\dot{\omega}_{2}\) of crank \(A D\) in Sample Problem 7/3 for the conditions cited. Also find the angular acceleration \(\dot{\omega}_{n}\) of link \(A
The motor housing and its bracket rotate about the \(Z\)-axis at the constant rate \(\Omega=3 \mathrm{rad} / \mathrm{s}\). The motor shaft and disk have a constant angular velocity of spin \(p=8
The bent plate has a mass of \(70 \mathrm{~kg}\) per square meter of surface area and revolves about the \(z\)-axis at the rate \(\omega=30 \mathrm{rad} / \mathrm{s}\). Determine (a) the angular
The two circular disks, each of mass \(m_{1}\), are connected by the curved bar bent into quarter-circular arcs and welded to the disks. The bar has a mass \(m_{2}\). The total mass of the assembly
The turbine rotor in a ship's power plant has a mass of \(1000 \mathrm{~kg}\), with center of mass at \(G\) and a radius of gyration of \(200 \mathrm{~mm}\). The rotor shaft is mounted in bearings
A proposed space station is closely approximated by four uniform spherical shells, each of mass \(m\) and radius \(r\). The mass of the connecting structure and internal equipment may be neglected as
A body weighing \(25 \mathrm{lb}\) is suspended from a spring of constant \(k=160\) \(\mathrm{lb} / \mathrm{ft}\). At time \(t=0\), it has a downward velocity of \(2 \mathrm{ft} / \mathrm{sec}\) as
The 8-kg body is moved \(0.2 \mathrm{~m}\) to the right of the equilibrium position and released from rest at time \(t=0\). Determine its displacement at time \(t=2 \mathrm{~s}\). The viscous damping
The two fixed counterrotating pulleys are driven at the same angular speed \(\omega_{0}\). A round bar is placed off center on the pulleys as shown. Determine the natural frequency of the resulting
A 50-kg instrument is supported by four springs, each of stiffness 7500 \(\mathrm{N} / \mathrm{m}\). If the instrument foundation undergoes harmonic motion given in meters by \(x_{B}=0.002 \cos 50
The spring attachment point \(B\) is given a horizontal motion \(x_{B}=\) \(b \cos \omega t\). Determine the critical driving frequency \(\omega_{c}\) for which the oscillations of the mass \(m\)
The 100-lb piston is supported by a spring of modulus \(k=200 \mathrm{lb} / \mathrm{in}\). A dashpot of damping coefficient \(c=85 \mathrm{lb}\)-sec/ft acts in parallel with the spring. A fluctuating
A simplified version of a pendulum used in impact tests is shown in the figure. Derive the equation of motion and determine the period for small oscillations about the pivot. The mass center \(G\) is
The uniform bar of mass \(m\) and length \(l\) is pivoted at its center. The spring of constant \(k\) at the left end is attached to a stationary surface, but the right-end spring, also of constant
Derive the equation of motion for the homogeneous circular cylinder, which rolls without slipping. If the cylinder mass is \(50 \mathrm{~kg}\), the cylinder radius \(0.5 \mathrm{~m}\), the spring
The small sphere of mass \(m\) is mounted on the light \(\operatorname{rod}\) pivoted at \(O\) and supported at end \(A\) by the vertical spring of stiffness \(k\). End \(A\) is displaced a small
Determine the natural frequency \(\omega_{n}\) of vertical vibration of the \(3-\mathrm{kg}\) collar to which are attached the two uniform \(1.2-\mathrm{kg}\) links, which may be treated as slender
If the 100-kg mass has a downward velocity of \(0.5 \mathrm{~m} / \mathrm{s}\) as it passes through its equilibrium position, calculate the magnitude \(a_{\max }\) of its maximum acceleration. Each
Salt water is being discharged into the atmosphere from the two \(30^{\circ}\) outlets at the total rate of \(30 \mathrm{~m}^{3} / \mathrm{min}\). Each of the discharge nozzles has a flow diameter of
In a test of the operation of a "cherry-picker" fire truck, the equipment is free to roll with its brakes released. For the position shown, the truck is observed to deflect the spring of stiffness
The military jet aircraft has a gross weight of \(24,000 \mathrm{lb}\) and is poised for takeoff with brakes set while the engine is revved up to maximum power. At this condition, air with a specific
At a bulk loading station, gravel leaves the hopper at the rate of \(220 \mathrm{lb} / \mathrm{sec}\) with a velocity of \(10 \mathrm{ft} / \mathrm{sec}\) in the direction shown and is deposited on
The diverter section of pipe between \(A\) and \(B\) is designed to allow the parallel pipes to clear an obstruction. The flange of the diverter is secured at \(C\) by a heavy bolt. The pipe carries
A flywheel rotating freely at \(1800 \mathrm{rev} / \mathrm{min}\) clockwise is subjected to a variable counterclockwise torque which is first applied at time \(t=\) 0 . The torque produces a
The pinion \(A\) of the hoist motor drives gear \(B\), which is attached to the hoisting drum. The load \(L\) is lifted from its rest position and acquires an upward velocity of \(3 \mathrm{ft} /
The right-angle bar rotates clockwise with an angular velocity which is decreasing at the rate of \(4 \mathrm{rad} / \mathrm{s}^{2}\). Write the vector expressions for the velocity and acceleration
A wheel of radius \(r\) rolls on a flat surface without slipping. Determine the angular motion of the wheel in terms of the linear motion of its center \(O\). Also determine the acceleration of a
The load \(L\) is being hoisted by the pulley-and-cable arrangement shown. Each cable is wrapped securely around its respective pulley so it does not slip. The two pulleys to which \(L\) is attached
Motion of the equilateral triangular plate \(A B C\) in its plane is controlled by the hydraulic cylinder \(D\). If the piston rod in the cylinder is moving upward at the constant rate of \(0.3
The wheel of radius \(r=300 \mathrm{~mm}\) rolls to the right without slipping and has a velocity \(v_{O}=3 \mathrm{~m} / \mathrm{s}\) of its center \(O\). Calculate the velocity of point \(A\) on
Crank \(C B\) oscillates about \(C\) through a limited arc, causing crank \(O A\) to oscillate about \(O\). When the linkage passes the position shown with \(C B\) horizontal and \(O A\) vertical,
The common configuration of a reciprocating engine is that of the slider-crank mechanism shown. If the crank \(O B\) has a clockwise rotational speed of \(1500 \mathrm{rev} / \mathrm{min}\),
The power screw turns at a speed which gives the threaded collar \(C\) a velocity of \(0.8 \mathrm{ft} / \mathrm{sec}\) vertically down. Determine the angular velocity of the slotted arm when
The wheel of Sample Problem 5/7, shown again here, rolls to the right without slipping, with its center \(O\) having a velocity \(v_{O}=3 \mathrm{~m} / \mathrm{s}\). Locate the instantaneous center
Arm \(O B\) of the linkage has a clockwise angular velocity of \(10 \mathrm{rad} / \mathrm{sec}\) in the position shown where \(\theta=45^{\circ}\). Determine the velocity of \(A\), the velocity of
The wheel of radius \(r\) rolls to the left without slipping and, at the instant considered, the center \(O\) has a velocity \(\mathbf{v}_{O}\) and an acceleration \(\mathbf{a}_{O}\) to the left.
The linkage of Sample Problem 5/8 is repeated here. Crank \(C B\) has a constant counterclockwise angular velocity of \(2 \mathrm{rad} / \mathrm{s}\) in the position shown during a short interval of
The slider-crank mechanism of Sample Problem 5/9 is repeated here. The crank \(O B\) has a constant clockwise angular speed of \(1500 \mathrm{rev} / \mathrm{min}\). For the instant when the crank
At the instant represented, the disk with the radial slot is rotating about \(O\) with a counterclockwise angular velocity of \(4 \mathrm{rad} / \mathrm{sec}\) which is decreasing at the rate of \(10
The pin \(A\) of the hinged link \(A C\) is confined to move in the rotating slot of link \(O D\). The angular velocity of \(O D\) is \(\omega=2 \mathrm{rad} / \mathrm{s}\) clockwise and is constant
For the conditions of Sample Problem 5/17, determine the angular acceleration of \(A C\) and the acceleration of \(A\) relative to the rotating slot in \(\operatorname{arm} O D\).Problem 5/17,The pin
Aircraft \(B\) has a constant speed of \(150 \mathrm{~m} / \mathrm{s}\) as it passes the bottom of a circular loop of 400-m radius. Aircraft \(A\) flying horizontally in the plane of the loop passes
Each of the three balls has a mass \(m\) and is welded to the rigid equiangular frame of negligible mass. The assembly rests on a smooth horizontal surface. If a force \(\mathbf{F}\) is suddenly
Consider the same conditions as for Sample Problem 4/2, except that the spokes are freely hinged at \(O\) and so do not constitute a rigid system. Explain the difference between the two
A shell with a mass of \(20 \mathrm{~kg}\) is fired from point \(O\), with a velocity \(u=\) \(300 \mathrm{~m} / \mathrm{s}\) in the vertical \(x-z\) plane at the inclination shown. When it reaches
The 32.2-lb carriage \(A\) moves horizontally in its guide with a speed of \(4 \mathrm{ft} / \mathrm{sec}\) and carries two assemblies of balls and light rods which rotate about a shaft at \(O\) in
The smooth vane shown diverts the open stream of fluid of crosssectional area \(A\), mass density \(ho\), and velocity \(v\). (a) Determine the force components \(R\) and \(F\) required to hold the
For the moving vane of Sample Problem 4/6, determine the optimum speed \(u\) of the vane for the generation of maximum power by the action of the fluid on the vane.Problem 4/6,The smooth vane shown
The offset nozzle has a discharge area \(A\) at \(B\) and an inlet area \(A_{0}\) at \(C\). A liquid enters the nozzle at a static gage pressure \(p\) through the fixed pipe and issues from the
An air-breathing jet aircraft of total mass \(m\) flying with a constant speed \(v\) consumes air at the mass rate \(m_{a}^{\prime}\) and exhausts burned gas at the mass rate \(m_{g}^{\prime}\) with

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