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engineering
engineering mechanics statics

Questions and Answers of Engineering Mechanics Statics

Tests of impact on the fixed crash dummy are conducted using the 300-lb ram that is released from rest at θ = 30°, and allowed to fall and strike the dummy at θ = 90°. If the coefficient of
The vertical shaft is rotating with an angular velocity of 3 rad/s when θ = 0°. If a force F is applied to the collar so that θ = 90°, determine the angular velocity of the shaft. Also, find the
The 10-lb block slides on the smooth surface when the corner D hits a stop block S. Determine the minimum velocity v the block should have which would allow it to tip over on its side and land in the
Determine the height h of the bumper of the pool table, so that when the pool ball of mass m strikes it, no frictional force will be developed between the ball and the table at A. Assume the bumper
The pendulum consists of a 15-kg solid ball and 6-kg rod. If it is released from rest when θ1 = 90°, determine the angle θ2 after the ball strikes the wall, rebounds, and the pendulum swings
The 4-lb rod AB is hanging in the vertical position. A 2-lb block, sliding on a smooth horizontal surface with a velocity of 12 ft/s, strikes the rod at its end B. Determine the velocity of the block
The hammer consists of a 10-kg solid cylinder C and 6-kg uniform slender rod AB. If the hammer is released from rest when θ = 90° and strikes the 30-kg block D when θ = 0º, determine the velocity
The 20-kg disk strikes the step without rebounding. Determine the largest angular velocity ω1 the disk can have without losing contact with the step A. 200 mm A 130 mm
The solid ball of mass m is dropped with a velocity v1 onto the edge of the rough step. If it rebounds horizontally off the step with a velocity v2, determine the angle θ at which contact
The uniform rod assembly rotates with an angular velocity of ω0 on the smooth horizontal plane just before the hook strikes the peg P without rebound. Determine the angular velocity of the assembly
The ladder of the fire truck rotates around the z axis with an angular velocity ω₁ = 0.15 rad/s, which is increasing at 0.8 rad/s². At the same instant it is rotating upward at a constant rate
A 50-g pencil (uniform slender rod) falls down onto the table with a velocity of 0.2 m/s just before impact. If it is at an angle of 60° with the horizontal, and it does not slip during the impact,
The rod of mass m and length L is released from rest without rotating. When it falls a distance L, the end A strikes the hook S, which provides a permanent connection. Determine the angular velocity
The 15-lb rod AB is released from rest in the vertical position. If the coefficient of restitution between the floor and the cushion at B is e = 0.7 determine how high the end of the rod rebounds
A ball having a mass of 8 kg and initial speed of v1 = 0.2 m/s rolls over a 30-mm-long depression.Assuming that the ball rolls off the edges of contact, first A then B, without slipping, determine
The 2-kg disk is thrown down onto the rough surface with the velocity and angular velocity shown. If there is no slipping, and the coefficient of restitution is e = 0.5, determine the velocity of the
The ladder of the fire truck rotates around the z axis with an angular velocity of ω₁ = 0.15 rad/s, which is increasing at 0.2 rad/s². At the same instant it is rotating upward at ω2 = 0.6 rad/s
The disk rotates about the z axis at a constant rate ωz = 0.5 rad/s without slipping on the horizontal plane. Determine the velocity and the acceleration of point A on the disk. x w₂ = 0.5
The cone rolls in a circle and rotates about the z axis at a constant rate ωz = 8 rad/s. Determine the angular velocity and angular acceleration of the cone if it rolls without slipping. Also, what
The pendulum consists of a 10-lb solid ball and 4-lb rod. If it is released from rest when θ = 0°, determine the angleθ₁ of rebound after the ball strikes the wall and thependulum swings up to
The propeller of an airplane is rotating at a constant speed ωxi, while the plane is undergoing a turn at a constant rate ωt. Determine the angular acceleration of the propeller if(a) The turn is
The antenna is following the motion of a jet plane. At the instant θ = 25° and ϕ = 75°, the constant angular rates of change are θ̇ = 0.4 rad/s and ϕ̇ = 0.6 rad/s. Determine the velocity and
The disk rotates about the shaft S, while the shaft is turning about the z axis at a rate of ωz = 4 rad/s, which is increasing at 2 rad/s2. Determine the velocity and acceleration of point B on the
At a given instant, the antenna has an angular motion ω₁ = 3 rad/s and ω̇₁ = 2 rad/s² about the z axis. At this same instant, θ = 30°, the angular motion about the x axis is ω₂ = 1.5
The disk rotates about the shaft S, while the shaft is turning about the z axis at a rate of ωz = 4 rad/s, which is increasing at 2 rad/s2. Determine the velocity and acceleration of point A on the
The telescope is mounted on the frame F that allows it to be directed to any point in the sky. At the instant θ = 30°, the frame has an angular acceleration of αy′ = 0.2 rad/s² and an angular
The bevel gear A rolls on the fixed gear B. If at the instant shown the shaft to which A is attached is rotating at 2 rad/s and has an angular acceleration of 4 rad/s2, determine the angular velocity
The truncated double cone rotates about the z axis at ωz = 0.4 rad/s without slipping on the horizontal plane. If at this same instant ωz is increasing at ω̇z = 0.5 rad/s², determine the
The electric fan is mounted on a swivel support such that the fan rotates about the z axis at a constant rate of ωz = 1 rad/s and the fan blade is spinning at a constant rate ωs = 60 rad/s. If ϕ =
The electric fan is mounted on a swivel support such that the fan rotates about the z axis at a constant rate of ωz = 1 rad/s and the fan blade is spinning at a constant rate ωs = 60 rad/s. If at
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 +
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
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
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 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
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
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.
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
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.
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
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
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
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
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
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
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,
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
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
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

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