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engineering
engineering mechanics dynamics

Questions and Answers of Engineering Mechanics Dynamics

The body consists of two slender uniform rods which have a mass per unit length. The rods are welded together and pivot about a horizontal axis through O against the action of a torsional spring of
By the method of this article, determine the period of vertical oscillation. Each spring has a stiffness of 6 lb/in., and the mass of the pulleys may be neglected. 50 lb
The uniform slender rod of length l and mass m2 is secured to the uniform disk of radius l/5 and mass m1. If the system is shown in its equilibrium position, determine the natural frequency wn and
Derive the natural frequency ƒn of the system composed of two homogeneous circular cylinders, each of mass M, and the connecting link AB of mass m. Assume small oscillations. M M ro B, To m LA
The rotational axis of the turntable is inclined at an angle a from the vertical. The turntable shaft pivots freely in bearings which are not shown. If a small block of mass m is placed a distance r
The assembly of Prob. 8/95 is repeated here without the applied harmonic torque. By the method of this article, determine the equation of motion of the system in terms of x for small oscillations
The ends of the uniform bar of mass m slide freely in the vertical and horizontal slots as shown. If the bar is in static equilibrium when = 0, determine the natural frequency wn of small
The 12-kg block is supported by the two 5-kg links with two torsion springs, each of constant kT = 500 N ∙ m/rad, arranged as shown. The springs are sufficiently stiff so that stable equilibrium is
The front-end suspension of an automobile is shown. Each of the coil springs has a stiffness of 270 lb/in. If the weight of the front-end frame and equivalent portion of the body attached to the
Determine the natural frequency ƒn of the inverted pendulum. Assume small oscillations, and note any restrictions on your solution. m k Aww k
Determine the period τ of small oscillations for the uniform sector of mass m = 1.0 kg, side b = 675 mm and angle β = 18°. The torsional spring of modulus kT = 8 N · m/rad is undeflected when the
The 0.1-kg projectile is fi red into the 10-kg block which is initially at rest with no force in the spring. The spring is attached at both ends. Calculate the maximum horizontal displacement X of
The uniform circular disk is suspended by a socket (not shown) which fits over the small ball attachment at O. Determine the frequency of small motion if the disk swings freely about (a) Axis
Determine the natural frequency ƒn for small oscillations of the semicircular slender rod about the equilibrium position if (a) r = 150 mm.(b) r = 300 mm.(c) r = 600 mm. Motion takes place
The triangular frame is constructed of uniform slender rod and pivots about a horizontal axis through point O. Determine the critical driving frequency wc of the block B which will result in
A linear oscillator with mass m, spring constant k, and viscous damping coefficient c is set into motion when released from a displaced position. Derive an expression for the energy loss Q during one
Calculate the damping ratio ζ of the system shown if the weight and radius of gyration of the stepped cylinder are W = 20 lb and k̅ = 5.5 in., the spring constant is k = 15 lb/in., and the damping
Determine the value of the viscous damping coefficient c for which the system is critically damped. The cylinder mass is m = 2 kg and the spring constant is k = 150 N/m. Neglect the mass and friction
The seismic instrument shown is secured to a ship’s deck near the stern where propeller-induced vibration is most pronounced. The ship has a single three-bladed propeller which turns at 180 rev/min
The square plate of side length 2b pivots about its center point O against the action of four springs, each of stiffness k. Each spring is attached to the corner of the plate at one end and to the
An experimental engine weighing 480 lb is mounted on a test stand with spring mounts at A and B, each with a stiffness of 600 lb/in. The radius of gyration of the engine about its mass center G is
The uniform bar of mass M and length l has a small roller of mass m with negligible bearing friction at each end. Determine the period ???? of the system for small oscillations on the curved track.
A 200-kg machine rests on four floor mounts, each of which has an effective spring constant k = 250 kN/m and an effective viscous damping coefficient c = 1000 N∙ s/m. The floor is known to vibrate
The mass of a critically damped system having a natural frequency wn = 4 rad/s is released from rest at an initial displacement x0. Determine the time t required for the mass to reach the position x
The uniform sector of Prob. 8/77 is repeated here with m = 4 kg, r = 325 mm, and β = 45°. If the sector is released from rest with θ0 = 90°, plot the value of for the time period 0 ≤ t ≤ 6
The mass of the system shown is released with the initial conditions x0 = 0.1 m and x˙0 = −5 m/s at t = 0. Plot the response of the system and determine the time(s) (if any) at which the
Shown in the figure are the elements of a displacement meter used to study the motion yB = b sin wt of the base. The motion of the mass relative to the frame is recorded on the rotating drum. If l1 =
The 4-kg mass is suspended by the spring of stiffness k = 350 N/m and is initially at rest in the equilibrium position. If a downward force F = Ct is applied to the body and reaches a value of 40 N
Plot the response x of the 50-lb body over the time interval 0 ≤ t ≤ 1 second. Determine the maximum and minimum values of x and their respective times. The initial conditions are x0 = 0 and x˙0
Determine and plot the response x as a function of time for the undamped linear oscillator subjected to the force F which varies linearly with time for the first 3/4 second as shown. The mass is
The 4-kg cylinder is attached to a viscous damper and to the spring of stiffness k = 800 N/m. If the cylinder is released from rest at time t = 0 from the position where it is displaced a distance y
Determine the mass moment of inertia of the bent uniform slender rod about the x- and y-axes shown, and about the z-axis. The rod has a mass ρ per unit length. y 2. /2
Determine the mass moment of inertia of the uniform thin triangular plate of mass m about the x-axis. Also determine the radius of gyration about the x-axis. By analogy state Iyy and ky. Then
Determine the mass moment of inertia about the y-axis for the uniform thin equilateral triangular plate of mass m. Also determine its radius of gyration about the y-axis. y b. 21
Calculate the mass moment of inertia of the homogeneous right-circular cone of mass m, base radius r, and altitude h about the cone axis x and about the y-axis through its vertex. y
Determine the mass moment of inertia of the uniform thin parabolic plate of mass m about the x-axis. State the corresponding radius of gyration. y 2 -Parabolic h m
For the thin homogeneous plate of uniform thickness t and mass m, determine the mass moments of inertia about the x'-, y'-, and z'-axes through the end of the plate at A. Refer to the results of
Determine the mass moment of inertia about the x-axis of the thin elliptical plate of mass m. y by a
Determine the mass moment of inertia of the homogeneous solid of revolution of mass m about the x-axis. y y = kxl.5 m
Determine the mass moment of inertia of the homogeneous solid of revolution of the previous problem about the y- and z-axes.
Determine the radius of gyration about the y-axis for the steel part shown in section. The part is formed by revolving one of the trapezoidal areas 360° around the y-axis. y -30 -30- 40 - -x 20- 10
Develop an expression for the mass moment of inertia of the homogeneous solid of revolution of mass m about the y-axis. a y = kzn b. m
Determine the mass moment of inertia about the x-axis of the solid spherical segment of mass m. y R 2. -x-
Determine the moment of inertia about the generating axis of a complete ring (torus) of mass m having a circular section with the dimensions shown in the sectional view. -R-
The plane area shown in the top portion of the figure is rotated 180° about the x-axis to form the body of revolution of mass m shown in the lower portion of the figure. Determine the mass moment of
Determine Iyy for the homogeneous body of revolution of the previous problem.
The thickness of the homogeneous triangular plate of mass m varies linearly with the distance from the vertex toward the base. The thickness a at the base is small compared with the other dimensions.
Determine the moment of inertia, about the generating axis, of the hollow circular tube of mass m obtained by revolving the thin ring shown in the sectional view completely around the generating
Determine the moments of inertia of the hemispherical shell with respect to the x- and z-axes. The mass of the shell is m, and its thickness is negligible compared with the radius r.
The partial solid of revolution is formed by revolving the shaded area in the x-z plane 90° about the z-axis. If the mass of the solid is m, determine its mass moment of inertia about the z-axis. 22
A shell of mass m is obtained by revolving the quarter-circular section about the z-axis. If the thickness of the shell is small compared with a and if r = a/3, determine the radius of gyration of
Determine the mass moment of inertia and corresponding radius of gyration of the thin homogeneous parabolic shell about the y-axis. The shell has dimensions r = 70 mm and h = 200 mm, and is made of
Every “slender” rod has a finite radius r. Refer to Table D/4 and derive an expression for the percentage error e which results if one neglects the radius r of a homogeneous solid cylindrical rod
The two small spheres of mass m each are connected by the light rigid rod which lies in the x-z plane. Determine the mass moments of inertia of the assembly about the x-, y-, and z-axes. y 7. m m
The rectangular metal plate has a mass of 15 kg. Compute its moment of inertia about the y-axis. What is the magnitude of the percentage error e introduced by using the approximate relation 1/3ml2
Determine Ixx for the cylinder with a centered circular hole. The mass of the body is m. r2
Determine the mass moment of inertia about the z-axis for the right-circular cylinder with a central longitudinal hole. 2r m 1/7
Determine the moment of inertia of the half-ring of mass m about its diametral axis a-a and about axis b-b through the midpoint of the arc normal to the plane of the ring. The radius of the circular
A 6-in. steel cube is cut along its diagonal plane. Calculate the moment of inertia of the resulting prism about the edge x-x. 6" 6" 6"
The uniform coiled spring weighs 4 lb. Approximate its moments of inertia about the x-, y-, and z-axes from the analogy to the properties of a cylindrical shell. y 5" 5"- 3"
Determine the length L of each of the slender rods of mass m/2 which must be centrally attached to the faces of the thin homogeneous disk of mass m in order to make the mass moments of inertia of the
A badminton racket is constructed of uniform slender rods bent into the shape shown. Neglect the strings and the built-up wooden grip and estimate the mass moment of inertia about the y-axis through
Calculate the moment of inertia of the steel control wheel, shown in section, about its central axis. There are eight spokes, each of which has a constant cross-sectional area of 200 mm2. What
The welded assembly is made of a uniform rod which weighs 0.370 lb per foot of length and the semicircular plate which weighs 8 lb per square foot. Determine the mass moments of inertia of the
The uniform rod of length 4b and mass m is bent into the shape shown. The diameter of the rod is small compared with its length. Determine the moments of inertia of the rod about the three coordinate
The welded assembly shown is made from a steel rod which weighs 0.455 lb per foot of length. Determine the mass moment of inertia of the assembly (a) about the y-axis and (b) about the z-axis. 4" -y
Calculate the moment of inertia of the solid steel semicylinder about the x-x axis and about the parallel x0-x0 axis. (See Table D/1 for the density of steel.) 100 mm Ox 60 mm 60 mm
The body is constructed of a uniform square plate, a uniform straight rod, a uniform quarter-circular rod, and a particle (negligible dimensions). If each part has the indicated mass, determine the
The clock pendulum consists of the slender rod of length l and mass m and the bob of mass 7m. Neglect the effects of the radius of the bob and determine IO in terms of the bob position x. Calculate
Determine the mass moments of inertia of the bracket about the x- and x′-axes. The bracket is made from thin plate of uniform thickness and has a mass of 0.35 kg per square meter of area. y 40 mm /
A square plate with a quarter-circular sector removed has a net mass m. Determine its moment of inertia about axis A-A normal to the plane of the plate. А a y. a a A
The welded assembly consists of a cylindrical shell with a closed semicircular end. The shell is made from sheet metal with a mass of 24 kg/m2, and the end is made from metal plate with a mass of 36
Determine the mass moment of inertia of the steel bracket about the z-axis which passes through the midline of the base. 2" 3" 1" 1" 1" 1.5" 1.5" 2.
The slender metal rods are welded together in the configuration shown. Each 6-in. segment weighs 0.30 lb. Compute the moment of inertia of the assembly about the y-axis. 6" 6" 6" 6" 6"
The welded assembly is formed from thin sheet metal having a mass of 19 kg/m2. Determine the mass moments of inertia for the assembly about the x- and y-axes. 150 mm 240 mm 80 mm
Determine Ixx for the cone frustum, which has base radii r1 and r2 and mass m. y r2 G
A preliminary design model to ensure rotational stability for a spacecraft consists of the cylindrical shell and the two square panels as shown. The shell and panels have the same thickness and
Determine the radius of gyration of the aluminum part about the z-axis. The hole in the upper surface is drilled completely through the part. 20. 25 30 5. 10 10- 75 50 -y Dimensions in millimeters
Compute the moment of inertia of the mallet about the O-O axis. The mass of the head is 0.8 kg, and the mass of the handle is 0.5 kg. 60 mm 60 mm 240 mm 60 mm 40 mm dia
Determine the products of inertia about the coordinate axes for the unit which consists of four small particles, each of mass m, connected by the light but rigid slender rods. 9- 9- m т x m
Determine the products of inertia about the coordinate axes for the unit which consists of three small spheres, each of mass m, connected by the light but rigid slender rods. m m | y m
Determine the product of inertia Ixy for the slender rod of mass m. y 1/2 1/2
Determine the products of inertia of the uniform slender rod of mass m about the coordinate axes shown. т h b
Determine the products of inertia about the coordinate axes for the thin square plate with two circular holes. The mass of the plate material per unit area is ρ. y 이4.6_4|614.6-4 |
Determine the products of inertia about the coordinate axes for the thin plate of mass m which has the shape of a circular sector of radius a and angle β as shown. y | m a
The homogeneous plate of Prob. B/7 is repeated here. Determine the product of inertia for the plate about the x-y axes. The plate has mass m and uniform thickness t. y x = ky? h Thickness t
Determine by direct integration the product of inertia of the thin homogeneous triangular plate of mass m about the x-y axes. Then, use the parallel-axis theorem to determine the product of inertia
The aluminium casting consists of a 6-in. cube with a 4-in. cubical recess. Calculate the products of inertia of the casting about the axes shown. 4" 6" 4" 6" 6
The S-shaped piece is formed from a rod of diameter d and bent into the two semicircular shapes. Determine the products of inertia for the rod, for which d is small compared with r. d トく
Determine the products of inertia about the coordinate axes for the assembly which consists of uniform slender rods. Each rod has a mass ρ per unit length. a a a a
Prove that the moment of inertia of the rigid assembly of three identical balls, each of mass m and radius r, has the same value for all axes through O. Neglect the mass of the connecting rods. m m -
The plane of the thin circular disk of mass m and radius r makes an angle β with the x-z plane. Determine the product of inertia of the disk with respect to the y-z plane. yー 1. G
The L-shaped piece is cut from steel plate having a mass per unit area of 160 kg/m2. Determine and plot the moment of inertia of the piece about axis A-A as a function of θ from θ = 0 to θ = 90°
Determine the moment of inertia I about axis O-M for the uniform slender rod bent into the shape shown. Plot I versus θ from θ = 0 to θ = 90° and determine the minimum value of I and the angle a
Determine the inertia tensor for the homogeneous thin plate about the x-, y-, and z-axes. The plate has a mass m and uniform thickness t. What is the minimum angle, measured from the x-axis, which
The assembly of three small spheres connected by light rigid bars of Prob. B/58 is repeated here. Determine the principal moments of inertia and the direction cosines associated with the axis of
The thin plate has a mass ρ per unit area and is formed into the shape shown. Determine the principal moments of inertia of the plate about axes through O. 26 -y b
The slender rod has a mass ρ per unit length and is formed into the shape shown. Determine the principal moments of inertia about axes through O and calculate the direction cosines of the axis of
The welded assembly is formed from uniform sheet metal with a mass of 32 kg/m2. Determine the principal mass moments of inertia for the assembly and the corresponding direction cosines for each
Derive the equation of motion for the system shown in terms of the displacement x. The masses are coupled through the light connecting rod ABC which pivots about the smooth bearing at point O.

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