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engineering
engineering mechanics statics

Engineering Mechanics Statics & Dynamics 15th Edition Russell C. Hibbeler - Solutions

The motion of the top is such that at the instant shown it rotates about the z axis at ω1 = 0.6 rad/s, while it spins at ω2 = 8 rad/s. Determine the angular velocity and angular acceleration of the top at this instant. Express the result as a Cartesian vector. X 001 Z 45° 002 -y
Gear C is driven by shaft DE, while gear B spins freely about its axle GF, which precesses freely about shaft DE. If gear A is held fixed (ωA = 0), and shaft DE rotates with a constant angular velocity of ωDE = 10 rad/s, determine the angular velocity of gear B. B G 150 mm Z D E -F WA C A 150
If the frame rotates with a constant angular velocity of ωp = {-10k} rad/s and the horizontal gear B rotates with a constant angular velocity of ωB = {5k} rad/s, determine the angular velocity and angular acceleration of the bevel gear A. Tallishishl B Z C wp A 1.5 ft- 0.75 ft
The truncated cone rotates about the z axis at a constant rate ωz = 0.4 rad/s without slipping on the horizontal plane. Determine the velocity and acceleration of point A on the cone. N Z. = 0.4 rad/s 0.5 ft 1 ft. 30⁰ 2 ft A 1.5 ft -y
Gear B is driven by a motor mounted on turntable C. If gear A is held fixed, and the motor shaft rotates with a constant angular velocity of ωy = 30 rad/s, determine the angular velocity and angular acceleration of gear B. A C Z -0.3 m- B 30 rad/s Wy "Gaism 0.15 m ↓
Shaft BD is connected to a ball-and-socket joint at B, and a beveled gear A is attached to its other end. The gear is in mesh with a fixed gear C. If the shaft and gear A are spinning with a constant angular velocity ω1 = 8 rad/s, determine the angular velocity and angular acceleration of gear A.
Gear B is driven by a motor mounted on turntable C. If gear A and the motor shaft rotate with constant angular speeds of ωA = {10k} rad/s and ωy = {30j} rad/s, respectively, determine the angular velocity and angular acceleration of gear B. A C Z -0.3 m B wy = 30 rad/s bol -y 0.15 m
The rod AB is attached to collars at its ends by ball-andsocket joints. If collar A has a velocity vA = 15 ft/s at the instant shown, determine the velocity of collar B. X 15 ft/s A 2 ft 2. 6 ft B 3 ft
The crane boom OA rotates about the z axis with a constant angular velocity of ω₁ = 0.15 rad/s, while it is rotating downward with a constant angular velocity of ω₂ = 0.2 rad/s. Determine the velocity and acceleration of point A located at the end of the boom at the instant shown. X 50
The differential of an automobile allows the two rear wheels to rotate at different speeds when the automobile travels along a curve. For operation, the rear axles are attached to the wheels at one end and have beveled gears A and B on their other ends. The differential case D is placed over the
Rod AB is attached to collars at its ends by using ball-andsocket joints. If collar A moves along the fixed rod with a velocity of vA = 5 m/s and has an acceleration aA = 2 m/s2 at the instant shown, determine the angular acceleration of the rod and the acceleration of collar B at this instant.
Rod AB is attached to the rotating arm using ball -andsocket joints. If AC is rotating about point C with an angular velocity of 8 rad/s and has an angular acceleration of 6 rad/s2 at the instant shown, determine the angular velocity and angular acceleration of link BD at this instant. X 6 ft 2
The rod AB is attached to collars at its ends by ball-andsocket joints. If collar A has an acceleration of aA = 2 ft/s2 at the instant shown, determine the acceleration of collar B. x 15 ft/s A 2 ft Z 6 ft B 3 ft
Rod AB is attached to collars at its ends by using ball- and-socket joints. If collar A moves along the fixed rod at vA = 5 m/s, determine the angular velocity of the rod and the velocity of collar B at the instant shown. Assume that the rod’s angular velocity is directed perpendicular to the
Rod AB is attached to collars at its ends by ball-and-socket joints. If collar A has a speed vA = 4 m/s, determine the speed of collar B at the instant z = 2 m. Assume the angular velocity of the rod is directed perpendicular to the rod. VA = 4 m/s X- Z A 2m 1.5 m N -1 m- B 1.5 m y
Rod CD is attached to the rotating arms using ball-andsocket joints. If AC has the motion shown, determine the angular velocity of link BD at the instant shown. X A Z @AC = 3 rad/s AC = 2 rad/s² 0.4 m C D 0.8 m B 0.6 m 1'm
Rod CD is attached to the rotating arms using ball-andsocket joints. If AC has the motion shown, determine the angular acceleration of link BD at the instant shown. A Z N— @AC = 3 rad/s AC = 2 rad/s² 0.4 m 0.8 m C D B 0.6 m 1'm
Rod AB is attached to the rotating arm using ball-andsocket joints. If AC is rotating with a constant angular velocity of 8 rad/s about the pin at C, determine the angular velocity of link BD at the instant shown. X 6 ft 2 ft D 1.5 ft it 3 ft B N JAC WAC = 8 rad/s
Solve Prob. 20–33 if the connection at B consists of a pin as shown in the figure below, rather than a ball-and-socket joint. The constraint allows rotation of the rod both along the bar (j direction) and along the axis of the pin (n direction). Since there is no rotational component in the u
If the rod is attached with ball-and-socket joints to smooth collars A and B at its end points, determine the velocity of B at the instant shown if A is moving upward at a constant speed of vA = 5 ft/s. Also, determine the angular velocity of the rod if it is directed perpendicular to the axis of
If the collar A in Prob. 20–33 is moving upward with an acceleration of aA = {-2k} ft/s2 at the instant its speed is vA = 5 ft/s, determine the acceleration of the collar at B at this instant. X- 2 ft B -6 ft- VA = 5 ft/s 3 'ft
Disk A rotates at a constant angular velocity of 10 rad/s. If rod BC is joined to the disk and a collar by ball-and-socket joints, determine the velocity of collar B at the instant shown. Also, what is the rod’s angular velocity ωBC if it is directed perpendicular to the axis of the rod? X 200
At the instant θ = 60°, the telescopic boom AB of theconstruction lift is rotating with a constant angular velocityabout the z axis of ω₁ = 0.5 rad/s and about the pin at Awith a constant angular speed of ω₂ = 0.25 rad/s.Simultaneously, the boom is extending with a velocity of1.5 ft/s, and
At the instant θ = 60°, the construction lift is rotating about the z axis with an angular velocity of ω₁ = 0.5 rad/s, and an angular acceleration of ω̇₁ = 0.25 rad/s² while the telescopic boom AB rotates about the pin at A with an angular velocity of ω₂ = 0.25 rad/s and
Solve Example 20.5 such that the x, y, z axes move with curvilinear translation, Ω = 0, in which case the collarappears to have both an angular velocity Ωxyz = ω₁+ ω2 and radial motion.Solve Example 20.5The pendulum shown in Fig. 20-13 consists of two rods; AB is pin supported at A and swings
Solve Example 20.5 by fixing x, y, z axes to rod BD so that Ω = ω₁ + ω₂. In this case the collar appears only to move radially outward along BD; hence Ωxyz = 0.Solve Example 20.5The pendulum shown in Fig. 20-13 consists of two rods; AB is pin supported at A and swings only in the Y-Z plane,
At the instant shown, the arm AB is rotating about the fixed pin A with an angular velocity ω₁ = 4 rad/s and an angular acceleration ω̇₁ = 3 rad/s². At the same instant, rod BD is rotating relative to rod AB with an angular velocity ω₂ = 5 rad/s which is increasing at ω̇₂ = 7
At a given instant the boom AB of the tower crane rotates about the z axis with the motion shown. At this same instant, θ = 60° and the boom is rotating downward such thatθ̇ = 0.4 rad/s and θ̈ = 0.6 rad/s². Determine the velocity and acceleration of the end of the boom A at this instant. The
At a given instant rod BD is rotating about the x axis with an angular velocity ωBD = 2 rad/s and an angular acceleration αBD = 5 rad/s2. Also, when θ = 45° link AC is rotating at θ̇ = 4 rad/s and θ̈ = 2 rad/s2. Determine the velocity and acceleration of point A on the link at this instant.
At the instant shown, the industrial manipulator is rotating about the z axis at ω1 = 5 rad/s, and about joint B at ω2 = 2 rad/s. Determine the velocity and acceleration of the grip A at this instant, when ϕ = 30°, θ = 45°, and r = 1.6 m. X 2. @r 1.2 m O2 B 가 A y
At the instant θ = 30°, the frame of the crane and the boom AB rotate with a constant angular velocity of ω1 = 1.5 rad/s and ω2 = 0.5 rad/s, respectively. Determine the velocity and acceleration of point B at this instant. w₁,w₁ Z G 1.5 m FA 12 m 10 W2, W₂ B
At the instant θ = 30°, the frame of the crane is rotating withan angular velocity of ω₁ = 1.5 rad/s and angularacceleration of ω̇₁ = 0.5 rad/s², while the boom AB rotateswith an angular velocity of ω̇₂ = 0.5 rad/s and angularacceleration of ω₂ = 0.25 rad/s². Determine the
At a given instant, the rod has the angular motions shown, while the collar C is moving down relative to the rod with a velocity of 6 ft/s and an acceleration of 2 ft/s2. Determine the collar’s velocity and acceleration at this instant. X Z 595 0.8 ft @₁ = 8 rad/s @₁ = 12 rad/s² 30⁰ 6
At the instant shown, the arm OA of the conveyor belt is rotating about the z axis with a constant angular velocity ω₁ = 6 rad/s, while at the same instant the arm is rotating upward at a constant rate ω₂ 4 rad/s. If the conveyor is running at a constant rate ṙ = 5 ft/s, determine the
At the instant shown, the industrial manipulator is rotating about the z axis at ω₁ = 5 rad/s, and ω̇₁ = 2 rad/s²; and about joint B at ω₂ = 2 rad/s and ω̇₂ = 3 rad/s². Determine the velocity and acceleration of the grip A at this instant, when ϕ = 30°, θ = 45°, and r = 1.6 m.
Determine the product of inertia Iyz of the composite plate assembly. The plates have a weight of 6 lb/ft2. -0.5 ft X 0.5 ft. Z 0.5 ft 10.25 0.25 ft 0.5 ft -y
At a given instant, the crane is moving along the track with a velocity vCD = 8 m/s and acceleration of 9 m/s². Simultaneously, it has the angular motions shown. If the trolley T is moving outwards along the boom AB with a relative speed of 3 m/s and relative acceleration of 5 m/s², determine the
Derive the Euler equations of motion for Ω ≠ ω, i.e., Eqs. 21–26.Eqs. 21–26. ΣΜ = 10, - ΙΩ,ω, + ΙΩ,ως ΣΜ = 1× - ΙΩ,ω. + ΙΩ,ω, ΣΜ. = Ιώ, - ΙΩ,ω. + 1,Ω,ω, (21-26)
At the moment of take off, the landing gear of an airplane is retracted with a constant angular velocity of ωp = 2 rad/s, while the wheel continues to spin. If the plane takes off with a speed of v = 320 km/h, determine the torque at A due to the gyroscopic effect.The wheel has a mass of 50 kg,
Derive the scalar form of the rotational equation of motion about the x axis if Ω ≠ ω and the moments and products of inertia of the body are constant with respect to time.
A spring is stretched 175 mm by an 8-kg block. If the block is displaced 100 mm downward from its equilibrium position and given a downward velocity of 1.50 m/s, determine the differential equation which describes the motion. Assume that positive displacement is downward. Also, determine the
A spring has a stiffness of 800 N/m. If a 2-kg block is attached to the spring, pushed 50 mm above its equilibrium position, and released from rest, determine the equation that describes the block’s motion. Assume that positive displacement is downward.
A spring is stretched 200 mm by a 15-kg block. If the block is displaced 100 mm downward from its equilibrium position and given a downward velocity of 0.75 m/s, determine the equation which describes the motion.What is the phase angle? Assume that positive displacement is downward.
When a 20-lb weight is suspended from a spring, the spring is stretched a distance of 4 in. Determine the natural frequency and the period of vibration for a 10-lb weight attached to the same spring.
When a 3-kg block is suspended from a spring, the spring is stretched a distance of 60 mm. Determine the natural frequency and the period of vibration for a 0.2-kg block attached to the same spring rather than a 3-kg block.
Show that the angular velocity of a body, in terms of Euler angles ϕ, θ, and ψ, can be expressed as ω =(ϕ̇ sin θ sin ψ + θ̇ cos ψ)i + (ϕ̇ sin θ cos ψ - θ̇ sin ψ)j + (ϕ̇ cos θ + ψ̇ )k, where i, j, and k are directed along the x, y, z axes as shown in Fig. 21-15d.
An 8-kg block is suspended from a spring having a stiffness k = 80 N/m. If the block is given an upward velocity of 0.4 m/s when it is 90 mm above its equilibrium position, determine the equation which describes the motion and the maximum upward displacement of the block measured from the
The uniform rod of mass m is supported by a pin at A and a spring at B. If B is given a small sideward displacement and released, determine the natural period of vibration. A B www. k L
The body of arbitrary shape has a mass m, mass center at G, and a radius of gyration about G of kG. If it is displaced a slight amount from its equilibrium position and released, determine the natural period of vibration. 0 G
The two identical gears each have a mass of m and a radius of gyration about their center of mass of k0. They are in mesh with the gear rack, which has a mass of M and is attached to a spring having a stiffness k. If the gear rack is displaced slightly horizontally, determine the natural period of
A 2-lb weight is suspended from a spring having a stiffness k = 2 lb/in. If the weight is pushed 1 in. upward from its equilibrium position and then released from rest, determine the equation which describes the motion. What is the amplitude and the natural frequency of the vibration? Assume that
The 3-kg target slides freely along the smooth horizontal guides BC and DE, which are ‘nested’ in springs that each have a stiffness of k = 9 kN/m. If a 60-g bullet is fired with a velocity of 900 m/s and embeds into the target, determine the amplitude and frequency of oscillation of the
The rod of mass m is supported by two cords, each having a length ∫ If the rod is given a slight rotation about a vertical axis through its center and released, determine the period of oscillation. 1
A pendulum has a 0.4-m-long cord and is given a tangential velocity of 0.2 m/s toward the vertical from a position θ = 0.3 rad. Determine the equation which describes the angular motion.
A 3-kg block is suspended from a spring having a stiffness of k = 200 N/m If the block is pushed 50 mm upward from its equilibrium position and then released from rest, determine the equation that describes the motion.What are the amplitude and the frequency of the vibration? Assume that positive
The 20-lb rectangular plate has a natural period of vibration τ = 0.3 s, as it oscillates around the axis of rod AB. Determine the torsional stiffness k, measured in lb · ft/rad, of the rod. Neglect the mass of the rod. 2 ft A k B 4 ft
The 15-kg block is suspended from two springs having a different stiffness and arranged a) Parallel to each otherb) As a series. If the natural periods of oscillation of the parallel system and series system are observed to be 0.5 s and 1.5 s, respectively, determine the spring stiffnesses k1
The 50-lb wheel has a radius of gyration about its mass center G of kG = 0.7 ft. Determine the frequency of vibration if it is displaced slightly from the equilibrium position and released. Assume no slipping. 0.4 ft 1 1.2 ft k = 18 lb/ft www-o
A block of mass m is suspended from two springs having a stiffness of k1 and k2, arranged a) Parallel to each other, andb) As a series. Determine the equivalent stiffness of a single spring with the same oscillation characteristics and the period of oscillation for each case.
Determine the natural frequency for small oscillations of the 10-lb sphere when the rod is displaced a slight distance and released. Neglect the size of the sphere and the mass of the rod.The spring has an unstretched length of 1 ft. 1 ft k=5 lb/ft 1 ft 1 ft
A uniform board is supported on two wheels which rotate in opposite directions at a constant angular speed. If the coefficient of kinetic friction between the wheels and board is μ, determine the frequency of vibration of the board if it is displaced slightly, a distance x from the midpoint
The 10-kg disk is pin connected at its mass center. Determine the natural period of vibration of the disk if the springs have sufficient tension in them to prevent the cord from slipping on the disk as it oscillates. Assume that the initial stretch in each spring is δO. /150 mm k = 80 N/m k = 80
The disk has a weight of 10 lb and rolls without slipping on the horizontal surface as it oscillates about its equilibrium position. If the disk is displaced, by rolling it counterclockwise 0.4 rad, determine the equation which describes its oscillatory motion when it is released. Initially, the
The bell has a mass of 375 kg, a center of mass at G, and a radius of gyration about point D of kD 0.4 m.The tongue consists of a slender rod attached to the inside of the bell at C. If an 8-kg mass is attached to the end of the rod, determine the length l of the rod so that the bell will “ring
If the springs in Prob. 22–24 are originally unstretched, determine the frequency of vibration. 150 mm k = 80 N/m k = 80 N/m
The block has a mass m and is supported by a rigid bar of negligible mass. If the spring has a stiffness k, determine the natural period of vibration for the block. A a -b-
Determine the frequency of vibration for the block. The springs are originally compressed Δ. k www k www m k www k www
The cylinder of radius r and mass m is displaced a small amount on the curved surface. If it rolls without slipping, determine the frequency of oscillation when it is released. R
The 25-lb weight is fixed to the end of the rod assembly. If both springs are unstretched when the assembly is in the position shown, determine the natural period of vibration for the weight when it is displaced slightly and released. Neglect the size of the block and the mass of the rods. k = 2
Determine the differential equation of motion of the block of mass m when it is displaced slightly and released. Motion occurs in the vertical plane. The springs are attached to the block. k₂
Determine the frequency of oscillation of the cylinder of mass m when it is pulled down slightly and released. Neglect the mass of the small pulley.
Determine the natural period of vibration of the 10-lb semicircular disk. 0.5 ft
A torsional spring of stiffness k is attached to a wheel that has a mass of M. If the wheel is given a small angular displacement of θ about the z axis, determine the natural period of oscillation. The wheel has a radius of gyration about the z axis of kz. 19
If the lower end of the 6-kg slender rod is displaced a small amount and released from rest, determine the natural frequency of vibration. Each spring has a stiffness of k = 200 N/m and is unstretched when the rod is hanging vertically. 2m WWW - k k 2 m
Determine the differential equation of motion of the 15-kg spool. Assume that it does not slip at the surface of contact as it oscillates. The radius of gyration of the spool about its center of mass is kG = 125 mm. The springs are originally unstretched. 200 mm k = 200 N/m 100 mm G k = 200 N/m
The 5-lb sphere is attached to a rod of negligible mass and rests in the horizontal position. Determine the natural frequency of vibration. Neglect the size of the sphere. -1 ft -0.5 ft- k = 10 lb/ft
Determine the natural period of vibration of the disk having a mass m and radius r. Assume the disk does not slip on the surface of contact as it oscillates. k wwww-of
Without an adjustable screw, A, the 1.5-lb pendulum has a center of gravity at G. If it is required that it oscillates with a period of 1 s, determine the distance a from pin O to the screw. The pendulum's radius of gyration about O is kO = 8.5 in. and the screw has a weight of 0.05 lb. a 7.5
If the block-and-spring model is subjected to the periodic force F = F0cos ot, show that the differential equation ofmotion is ẍ + (k/m)x= (F0/m) cos ωt, where x ismeasured from the equilibrium position of the block. Whatis the general solution of this equation?
The bar has a mass of 8 kg and is suspended from two springs such that when it is in equilibrium, the springs make an angle of 45° with the horizontal as shown. Determine the natural period of vibration if the bar is pulled down a short distance and released. Each spring has a stiffness of k = 40
If the wheel is given a small angular displacement of θ and released from rest, it is observed that it oscillates with a natural period of τ. Determine the wheel’s radius of gyration about its center of mass G. The wheel has a mass of m and rolls on the rails without slipping. 0 R
A block which has a mass m is suspended from a spring having a stiffness k. If an impressed downward vertical force F = FO acts on the weight, determine the equation which describes the position of the block as a function of time.
A 5-kg block is suspended from a spring having a stiffness of 300 N/m. If the block is acted upon by a vertical force F = (7 sin 8t) N, where t is in seconds, determine the equation which describes the motion of the block when it is pulled down 100 mm from the equilibrium position and released from
If the dashpot has a damping coefficient of c = 50 N · s/m, and the spring has a stiffness of k = 600 N/m, show that the system is underdamped, and then find the pendulum's period of oscillation. The uniform rods have a mass per unit length of 10 kg/m. A -0.3 m c = 50 N-s/m C D -0.3 m B k = 600
A 4-lb weight is attached to a spring having a stiffness k = 10 lb/ft. The weight is drawn downward a distance of 4 in. and released from rest. If the support moves with a vertical displacement δ = (0.5 sin 4t) in., where t is in seconds, determine the equation which describes the position of the
The 30-lb block is attached to two springs having a stiffness of 10 lb/ft. A periodic force F = (8 cos 3t) lb, where t is in seconds, is applied to the block. Determine the maximum speed of the block after frictional forces cause the free vibrations to dampen out. k = 10 lb/ft -www- k = 10
A 4-kg block is suspended from a spring that has a stiffness of k = 600 N/m. The block is drawn downward 50 mm from the equilibrium position and released from rest when t = 0. If the support moves with an impressed displacement of δ = (10 sin 4t) mm, where t is in seconds, determine the equation
Find the differential equation for small oscillations in terms of θ for the uniform rod of mass m. Also show that if c < √mk/2, then the system remains underdamped. The rod is in a horizontal position when it is in equilibrium. A B 2a C C
The barrel of a cannon has a mass of 700 kg, and after firing it recoils a distance of 0.64 m. If it returns to its original position by means of a single recuperator having a damping coefficient of 2 kN · s/m, determine the required stiffness of each of the two springs fixed to the base and
The light elastic rod supports a 4-kg sphere. When an 18-N vertical force is applied to the sphere, the rod deflects 14 mm. If the wall oscillates with harmonic frequency of 2 Hz and has an amplitude of 15 mm, determine the amplitude of vibration for the sphere. -0.75 m-
If the 30-kg block is subjected to a periodic force of P = (300 sin 5t) N, k = 1500 N/m, and c = 300 N · s/m, determine the equation that describes the steady-state vibration as a function of time. k www Vov с P = (300 sin 5 t)N
The fan has a mass of 25 kg and is fixed to the end of a horizontal beam that has a negligible mass.The fan blade is mounted eccentrically on the shaft such that it is equivalent to an unbalanced 3.5-kg mass located 100 mm from the axis of rotation. If the static deflection of the beam is 50 mm as
Use a block-and-spring model like that shown in Fig. 14a but suspended from a vertical position and subjected to a periodic support displacement of δ = δ0 cos ω0t determine the equation of motion for the system, and obtain its general solution. Define the displacement y measured from the static
The engine is mounted on a foundation block which is spring supported. Describe the steady-state vibration if the block and engine have a total weight of 1500 lb, and the engine, when running, creates an impressed force F = (50 sin 2t) lb, where t is in seconds. Assume that the system vibrates only
In Prob. 22–53, determine the amplitude of steady-state vibration of the fan if its angular velocity is 10 rad/s.Prob. 22–53The fan has a mass of 25 kg and is fixed to the end of a horizontal beam that has a negligible mass.The fan blade is mounted eccentrically on the shaft such that it is
The electric motor turns an eccentric flywheel which is equivalent to an unbalanced 0.25-lb weight located 10 in. from the axis of rotation. If the static deflection of the beam is 1 in. because of the weight of the motor, determine the angular velocity of the flywheel at which resonance will
What will be the amplitude of steady-state vibration of the fan in Prob. 22-53 if the angular velocity of the fan blade is 18 rad/s?Prob. 22-53The fan has a mass of 25 kg and is fixed to the end of a horizontal beam that has a negligible mass.The fan blade is mounted eccentrically on the shaft such
The 450-kg trailer is pulled with a constant speed over the surface of a bumpy road, which may be approximated by a cosine curve having an amplitude of 50 mm and wave length of 4 m. If the two springs s which support the trailer each have a stiffness of 800 N/m, determine the speed which will cause
The motor of mass M is supported by a simply supported beam of negligible mass. If block A of mass m is clipped onto the rotor, which is turning at constant angular velocity of , determine the amplitude of the steady-state vibration. When the beam is subjected to a concentrated force of P at its
Determine the angular velocity of the flywheel in Prob. 22–57 which will produce an amplitude of vibration of 0.25 in.Prob. 22–57The electric motor turns an eccentric flywheel which is equivalent to an unbalanced 0.25-lb weight located 10 in. from the axis of rotation. If the static deflection

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