Concept explainers
The top shown is supported at the fixed point O. Denoting by ϕ, θ, and ψ the Eulerian angles defining the position of the top with respect to a fixed frame of reference, consider the general motion of the top in which all Eulerian angles vary.
(a) Observing that ΣMZ = 0 and ΣMz = 0, and denoting by I and I′, respectively, the moments of inertia of the top about its axis of symmetry and about a transverse axis through O, derive the two first-order differential equations of motion
where α and β are constants depending upon the initial conditions.
These equations express that the angular momentum of the top is conserved about both the Z and z axes; that is, that the rectangular component of HO along each of these axes is constant.
(b) Use Eqs. (1) and (2) to show that the rectangular component ωz of the angular velocity of the top is constant and that the rate of precession
Fig. P18.137 and P18.138
Want to see the full answer?
Check out a sample textbook solutionChapter 18 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
- A 350-kg machine is placed at the end of 1.8-m-long steel ( E = 210 x 109 N/m2)cantilever beam. The machine is observed to vibrate with a natural frequency of 35Hz.What is the moment of inertia of the beam's cross section about its neutral axis?arrow_forward(a) Show that the polar radius of gyration rO of the annular area shown is approximately equal to the mean radius Rm = (R1 + R2)/2 for small values of the thickness t = R2 – R1. (b) Determine the percentage error introduced by using Rm in place of rO for the following values of t/Rm: 1, 1/2, and 1/10.arrow_forwardA homogeneous cone with a height of h = 12 in. and a base with a diameter of d = 6 in. is attached as shown to a cord AB. Knowing that the angles that cord AB and the axis BC β=45 = 45° and 0= 30° and that the cone precesses at the constant rate O=8rd/s in the sense indicated, determine (a ) the rate of spin of the cone about its axis BC, (b ) the length of cord AB,arrow_forward
- A uniform thin disk with a 6-in. diameter is attached to the end of a rod AB of negligible mass that is supported by a ball-and-socket joint at point A. Knowing that the disk is spinning about its axis of symmetry AB at the rate of 2100 rpm in the sense indicated and that AB forms an angle β = 45° with the vertical axis AC, determine the two possible rates of steady precession of the disk about the axis AC.arrow_forwardThe angular velocity vector of a football that has just been kicked is horizontal, and its axis of symmetry OC is oriented as shown. Knowing that the magnitude of the angular velocity is 200 rpm and that the ratio of the axis and transverse moments of inertia is determine (a) the orientation of the axis of precession OA (a) the rates of precession and spin.arrow_forwardA truck with four wheels, each 750 mm diameter, travels on rails round a curve of 75 m at a speed of 50 km/h. The total mass of the truck is 5 t and its centre of gravity is midway between the axles, 1.05 m above the rails and midway between them. Each pair of wheels is driven by a motor rotating in the opposite direction to the wheels and at four times the speed. The moment of inertia of each pair of wheels is 15 Kgm2. The rails lie on a horizontal plane and 1.45 m apart. Determine the load on each rail.arrow_forward
- A uniform thin disk with a 6-in. diameter is attached to the end of a rod AB of negligible mass that is supported by a ball-and-socket joint at point A. Knowing that the disk is observed to precess about the vertical axis AC at the constant rate of 36 rpm in the sense indicated and that its axis of symmetry AB forms an angle β= 60° with AC, determine the rate at which the disk spins about rod AB.arrow_forwardii. A wheel is mounted on a shaft (G=83 x 109 N/m2) of length 1.5m and radius 0.8cm. The wheel is rotated 50 and released. The period of oscillation is observed as 2.3s. determine the mass moment of inertia of the wheel.arrow_forwardA yoyo is constructed by attaching three uniform, solid disks along their central axes as shown. The two outer disks are identical, each with mass M = 58 g, radius R = 3.3 cm, and moment of inertia 1/2MR2. The central, smaller disk has mass M/2 and radius R/2. A light, flexible string of negligible mass is wrapped counterclockwise around the central disk of the yoyo. The yoyo is then placed on a horizontal tabletop and the string is gently pulled with a constant force F = 0.25 N. The tension in the string is not sufficient to cause the yoyo to leave the tabletop. In this problem consider the two cases show. In Case 1 the string is pulled straight up, perpendicular to the tabletop. In Case 2 the string is pulled horizontally, parallel to the tabletop. In both cases the yoyo rolls without slipping. In both the cases shown what is the magnitude of the tourqe t excerted by the string about the contact point of the yo-yo wiith the table in N*m. What is the moment of intertia of the yo-yo…arrow_forward
- A solid cube of side c= 120 mm is attached as shown to a cord AB of length 240 mm. The cube spins about its diagonal BC and precesses about the vertical axis AD Knowing that 0= 25° and β= 40°, determine (a) the rate of spin of the cube, (b) its rate of precession. (See hint of Prob. 18.115.)arrow_forwardThe inner rim of an 85-lb flywheel is placed on a knife edge, and the period of its small oscillations is found to be 1.26 s. Determine the centroidal moment of inertia of the flywheel.arrow_forwardA coin is tossed into the air. It is observed to spin at the rate of 600 rpm about an axis GC perpendicular to the coin and to precess about the vertical direction GD Knowing that GC forms an angle of 15° with GD, determine (a) the angle that the angular velocity w of the coin forms with G (b) the rate of precession of the coin about GD.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY