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engineering
engineering mechanics dynamics

Questions and Answers of Engineering Mechanics Dynamics

The mechanism of a machine is constructed so that for a short time the roller at A follows the surface of the cam described by the equation r = a + b cosθ. If θ' and θ'' are given, determine the
A car is traveling along the circular curve having a radius r. At the instance shown, its angular rate of rotation is θ', which is decreasing at the rate θ''. Determine the radial and transverse
The small washer is sliding down the cord OA. When it is at the midpoint, its speed is v and its acceleration is a'. Express the velocity and acceleration of the washer at this point in terms of its
The roller coaster is traveling down along the spiral ramp with a constant speed v. If the track descends a distance h for every full revolution, determine the magnitude of the roller coaster’s
The motor draws in the cable at C with a constant velocity vC. The motor draws in the cable at D with a constant acceleration of aD. If vD = 0 when t = 0, determine(a) The time needed for block A to
For a short time the position of the roller-coaster car along its path is defined by the equations r = r0, θ = at, and z = bcos θ. Determine the magnitude of the car’s velocity and acceleration
The cord is attached to the pin at C and passes over the two pulleys at A and D. The pulley at A is attached to the smooth collar that travels along the vertical rod. Determine the velocity and
A cameraman standing at A is following the movement of a race car, B, which is traveling around a curved track at constant speed vB. Determine the angular rate at which the man must turn in order to
A double collar C is pin-connected together such that one collar slides over a fixed rod and the other slides over a rotating rod. If the geometry of the fixed rod for a short distance can be defined
The cord of length L is attached to the pin at C and passes over the two pulleys at A and D. The pulley at A is attached to the smooth collar that travels along the vertical rod. When sB = b, the end
The crate C is being lifted by moving the roller at A downward with constant speed vA along the guide. Determine the velocity and acceleration of the crate at the instant s = s1. When the roller is
The man pulls the boy up to the tree limb C by walking backward. If he starts from rest when xA = 0 and moves backward with constant acceleration aA, determine the speed of the boy at the instant yB
The girl at C stands near the edge of the pier and pulls in the rope horizontally at constant speed vC. Determine how fast the boat approaches the pier at the instant the rope length AB is d. Given:
At the instant shown, cars A and B are traveling at speeds vA and vB respectively. If B is increasing its speed at v'A, while A maintains a constant speed, determine the velocity and acceleration of
Cars A and B are traveling around the circular race track. At the instant shown, A has speed vA and is increasing its speed at the rate of v'A, whereas B has speed vB and is decreasing its speed at
At the instant shown, cars A and B are traveling at speeds vA and vB respectively. If A is increasing its speed at v'A whereas the speed of B is decreasing at v'B, determine the velocity and
Two planes, A and B, are flying at the same altitude. If their velocities are vA and vB such that he angle between their straight-line courses is θ, determine the velocity of plane B with respect to
An aircraft carrier is traveling forward with a velocity v0. At the instant shown, the plane at A has just taken off and has attained a forward horizontal air speed vA, measured from still water. If
The boy A is moving in a straight line away from the building at a constant speed vA. At what horizontal distance d must he be from C in order to make the catch if the ball is thrown with a
Two boats leave the shore at the same time and travel in the directions shown with the given speeds. Determine the speed of boat A with respect to boat B. How long after leaving the shore will the
The airplane has a speed relative to the wind of vA. If the speed of the wind relative to the ground is vW, determine the angle θ at which the plane must be directed in order to travel in the
At the instant shown, the car at A is traveling at vA around the curve while increasing its speed at v'A. The car at B is traveling at vB along the straightaway and increasing its speed at v'B.
The boy A is moving in a straight line away from the building at a constant speed vA. The boy C throws the ball B horizontally when A is at d. At what speed must C throw the ball so that A can catch
At the instant shown car A is traveling with a velocity vA and has an acceleration aA along the highway. At the same instant B is traveling on the trumpet interchange curve with a speed vB which is
At a given instant, two particles A and B are moving with a speed of v0 along the paths shown. If B is decelerating at v'B and the speed of A is increasing at v'A, determine the acceleration of A
Repeat Prob. 6/215 for the case of a slowly increasing forward acceleration a. Determine the ratio h/b which will allow the package to tip backward about the rear feet before it slides backward
The uniform 40-lb bar with attached 12-lb wheels is released from rest in the orientation shown. The wheels have a centroidal radius of gyration of 4.5 in., and the coefficients of static and kinetic
A device to produce vibrations consists of the two counter-rotating wheels, each carrying an eccentric mass m0 = 1 kg with a center of mass at a distance e = 12 mm from its axis of rotation. The
For the square pyramid of Prob. B/20, determine the mass moment of inertia about the x-axis. h -y Problem B/20 0/2 0/2
The seismic instrument shown is attached to a structure which has a horizontal harmonic vibration at 3 Hz. The instrument has a mass m = 0.5 kg, a spring stiffness k = 20 N/m, and a viscous damping
Determine the mass moment of inertia about the y-axis for the parabolic plate of the previous problem. State the radius of gyration about the y-axis. y 2 2 -Parabolic h -x- Problem B/5
If the triangular frame of Prob. 8/83 is made to oscillate about axis B-B, determine the natural circular frequency wn for small oscillations. Evaluate for l = 200 mm and compare your answer with
The 20-lb spoked wheel has a centroidal radius of gyration k = 6 in. A torsional spring of constant kT = 160 lb-ft/rad resists rotation about the smooth bearing. If an external torque of form M = M0
The mass of the uniform slender rod is 3 kg. Determine the position x for the 1.2-kg slider such that the system period is 1 s. Assume small oscillations about the horizontal equilibrium position
The cart B is given the harmonic displacement xB = b sin wt. Determine the steady-state amplitude Θ of the periodic oscillation of the uniform slender bar which is pinned to the cart at P. Assume
The system shown features a nonlinear spring whose resisting force F increases with deflection from the neutral position according to the graph shown. Determine the equation of motion for the system
For the steel part of Prob. B/11, determine the radius of gyration about the x-axis. y K-30 30– 40 20> 10 Dimensions in millimeters Problem B/11
Determine the moment of inertia about the y-axis for the paraboloid of revolution of Prob. B/13. h -y Problem B/13
Determine the radius of gyration about the z-axis of the paraboloid of revolution shown. The mass of the homogeneous body is m. h - y
For the partial solid of revolution in Prob. B/25, determine the mass moment of inertia about the x-axis. x = Problem B/25
Determine the mass moment of inertia of the homogeneous square pyramid about the z-axis. The pyramid has mass m. b. -y 2. b. -b-
For the parabolic shell of Prob. B/28, determine the mass moment of inertia and corresponding radius of gyration about the z-axis. h' Problem B/28
The rod of Prob. B/1 is repeated here. Determine the product of inertia for the rod about the x-y axes. The rod has a mass ρ per unit length.
Determine the products of inertia of the solid homogeneous half-cylinder of mass m for the axes shown.
The equilibrium position of the mass m occurs where y = 0 and yB = 0. When the attachment B is given a steady vertical motion yB = b sin wt, the mass m will acquire a steady vertical oscillation.
Attachment B is given a horizontal motion xB = b cos wt. Derive the equation of motion for the mass m and state the critical frequency wc for which the oscillations of the mass become
Attachment B is given a horizontal motion xB = b cos wt. Derive the equation of motion for the mass m and state the critical frequency wc for which the oscillations of the mass become excessively
Derive the equation of motion for the inertial displacement xi of the mass of Fig. 8/14. Comment on, but do not carry out, the solution to the equation of motion. k W ww k Problem 8/14
The instrument shown has a mass of 43 kg and is spring-mounted to the horizontal base. If the amplitude of vertical vibration of the base is 0.10 mm, calculate the range of frequencies ƒn of the
When the person stands in the center of the floor system shown, he causes a static deflection δst of the floor under his feet. If he walks (or runs quickly!) in the same area, how many steps per
A single-cylinder four-stroke gasoline engine with a mass of 90 kg is mounted on four stiff spring pads, each with a stiffness of 30(103) kN/m, and is designed to run at 3600 rev/min. The mounting
The 20-kg variable-speed motorized unit is restrained in the horizontal direction by two springs, each of which has a stiffness of 2.1 kN/m. Each of the two dashpots has a viscous damping coefficient
The motion of the outer frame B is given by xB = b sin wt. For what range of the driving frequency w is the amplitude of the motion of the mass m relative to the frame less than 2b? XB = b sin ot k/2
It was noted in the text that the maxima of the curves for the magnification factor M are not located at w/wn = 1. Determine an expression in terms of the damping ratio ζ for the frequency ratio at
The 4-lb body is attached to two springs, each of which has a stiffness of 6 lb/in. The body is mounted on a shake table which vibrates harmonically in the horizontal direction with an amplitude of
A viscously damped spring-mass system is forced harmonically at the undamped natural frequency (w/wn = 1). If the damping ratio ζ is doubled from 0.1 to 0.2, compute the percentage reduction R1 in
An external force F = F0 sin wt is applied to the cylinder as shown. What value wc of the driving frequency would cause excessively large oscillations of the system? k 2m m YF
The block of weight W = 100 lb is suspended by two springs each of stiffness k = 200 lb/ft and is acted upon by the force F = 75 cos 15t lb where t is the time in seconds. Determine the amplitude X
A spring-mounted machine with a mass of 24 kg is observed to vibrate harmonically in the vertical direction with an amplitude of 0.30 mm under the action of a vertical force which varies harmonically
Determine the angular acceleration a of the dumbbell of Prob. 7/14 for the conditions stated, except that Ω is increasing at the rate of 3 rad/s2 for the instant under consideration. т @p G m Y. X-
Determine the amplitude X of the steady-state motion of the 10-kg mass if (a) c = 500 N∙s/m and (b) c = 0. k = 100 kN/m m = 10 kg F = 1000 cos 120t N %3D
The solid circular cylinder of mass m, radius r, and length b revolves about its geometric axis at an angular rate p rad/s. Simultaneously, the bracket and attached shaft revolve about the x-axis at
In manipulating the dumbbell, the jaws of the robotic device have an angular velocity wp = 2 rad/s about the axis OG with γ fixed at 60°. The entire assembly rotates about the vertical Z-axis at
The spool A rotates about its axis with an angular velocity of 20 rad/s, first in the sense of wa and second in the sense of wb. Simultaneously, the assembly rotates about the vertical axis with an
If the motor of Sample Problem 7/2, repeated in Prob. 7/11, reaches a speed of 3000 rev/min in 2 seconds from rest with constant acceleration, determine the total angular acceleration of the rotor
The motor of Sample Problem 7/2 is shown again here. If the motor pivots about the x-axis at the constant rate ϒ˙ = 3π rad/sec with no rotation about the Z-axis (N = 0), determine the angular
The panel assembly and attached x-y-z axes rotate with a constant angular velocity Ω = 0.6 rad/sec about the vertical z-axis. Simultaneously, the panels rotate about the y-axis as shown with a
The rod is hinged about the axis O-O of the clevis, which is attached to the end of the vertical shaft. The shaft rotates with a constant angular velocity w0 as shown. If θ is decreasing at the
A slender rod bent into the shape shown rotates about the fixed line CD at a constant angular rate w. Determine the velocity and acceleration of point A. D y d c/ CA A
The rotor B spins about its inclined axis OA at the angular speed N1 = 200 rev/min, where β = 30°. Simultaneously, the assembly rotates about the vertical z-axis at the rate N2. If the total
The disk rotates with a spin velocity of 15 rad/s about its horizontal z-axis first in the direction (a) and second in the direction (b). The assembly rotates with the angular velocity N = 10 rad/s
The rotor and shaft are mounted in a clevis which can rotate about the z-axis with an angular velocity Ω. With Ω = 0 and θ constant, the rotor has an angular velocity w0 = −4j − 3k rad/s. Find
A timing mechanism consists of the rotating distributor arm AB and the fixed contact C. If the arm rotates about the fixed axis OA with a constant angular velocity w = 30(3i + 2j + 6k) rad/s, and if
The solid cylinder is rotating about the fixed axis OA with a constant speed N = 600 rev/min in the direction shown. If the x-and y-components of the velocity of point P are 12 ft/sec and −6
Repeat the experiment of Prob. 7/1 but use a small angle of rotation, say, 5°. Note the near-equal final positions for the two different rotation sequences. What does this observation lead you to
Reconsider the basic mechanism of Prob. 6/232, only now the mass of the crank OA is 1.2 kg and that of the uniform output arm BC is 1.8 kg. For simplicity, treat the crank OA as uniform. Determine
Place your textbook on your desk, with fixed axes oriented as shown. Rotate the book about the x-axis through a 90° angle and then from this new position rotate it 90° about the y-axis. Sketch the
Reconsider the mechanism of Prob. 6/232. If crank OA now starts from rest and acquires a speed of 60 rev/min in one complete revolution with constant angular acceleration, determine and plot the
The four-bar mechanism of Prob. 6/108 is repeated here. The coupler AB has a mass of 7 kg, and the masses of crank OA and the output arm BC may be neglected. Determine and plot the torque M which
The 60-ft telephone pole of essentially uniform diameter is being hoisted into the vertical position by two cables attached at B as shown. The end O rests on a fixed support and cannot slip. When the
For a train traveling at 160 km/h around a horizontal curve of radius 1.9 km, calculate the elevation angle β of the track so that passengers will feel only a force normal to their seats and the
The uniform 100-kg beam AB is hanging initially at rest with θ = 0 when the constant force P = 300 N is applied to the cable. Determine (a) The maximum angular velocity reached by the beam with
The compound pendulum is composed of a uniform slender rod of length l and mass 2m to which is fastened a uniform disk of diameter l/2 and mass m. The body pivots freely about a horizontal axis
The uniform power pole of mass m and length L is hoisted into a vertical position with its lower end supported by a fixed pivot at O. The guy wires supporting the pole are accidentally released, and
The steel I-beam is to be transported by the overhead trolley to which it is hinged at O. If the trolley starts from rest with θ = θ˙ = 0 and is given a constant horizontal acceleration a = 2
If the 1.2-kg uniform slender bar is released from rest in the position θ = 0 where the spring is unstretched, determine and plot its angular velocity as a function of θ over the range 0 ≤ θ ≤
The four-bar mechanism operates in a horizontal plane. At the instant illustrated, θ = 30° and crank OA has a constant counterclockwise angular velocity of 3 rad/s. Determine the required magnitude
The 6-lb pendulum with mass center at G is pivoted at A to the fixed support CA. It has a radius of gyration of 17 in. about O-O and swings through an amplitude θ = 60°. For the instant when the
Repeat the analysis of Prob. 6 /108 with the added information that the mass of crank OA is 1.2 kg and the mass of the output arm BC is 1.8 kg. Each of these bars may be considered uniform for this
For the beam described in Prob. 6/62, determine the maximum angular velocity w reached by the beam as it rotates in the vertical plane about the bearing at O. Also determine the corresponding force R
The slender rod of mass m1 and length L has a movable slider of mass m2 which can be tightened at any location x along the rod. The assembly is initially falling in translation with speed v1. A small
The uniform semicircular plate is at rest on the smooth horizontal surface when the force F is applied at B. Determine the coordinates of the point P in the plate which has zero initial acceleration.
A space telescope is shown in the figure. One of the reaction wheels of its attitude-control system is spinning as shown at 10 rad /s, and at this speed the friction in the wheel bearing causes an
The forklift truck with center of mass at G1 has a weight of 3200 lb including the vertical mast. The fork and load have a combined weight of 1800 lb with center of mass at G2. The roller guide at B
The small block of mass m slides along the radial slot of the disk while the disk rotates in the horizontal plane about its center O. The block is released from rest relative to the disk and moves
The body of mass m is supported by feet of negligible size and is at rest relative to the tilted surface, which is contained within an experimental vehicle. If the vehicle is brought to rest with a
The uniform slender bar weighs 60 lb and is released from rest in the near-vertical position shown, where the spring of stiffness 10 lb/ft is unstretched. Calculate the speed with which end A strikes
The gear train shown operates in a horizontal plane at a steady speed and receives 6 hp from a motor at A to move rack D against a 4500-pound load L. At what speed will the rack move if gears A, B,
Four identical slender rods each of mass m are welded at their ends to form a square, and the corners are then welded to a light metal hoop of radius r. If the rigid assembly of rods and hoop is
The dump truck carries 5 m3 of dirt with a density of 1600 kg/m3, and the elevating mechanism rotates the dump about the pivot A at a constant angular rate of 4 deg/s. The mass center of the dump and

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