Concept explainers
If a body contains no planes of symmetry, the principal moments of inertia can be determined mathematically. To show how this is done, consider the rigid body which is spinning with an angular velocity ω, directed along one of its principal axes of inertia. If the principal moment of inertia about this axis is I, the angular momentum can be expressed as H = Iω = Iωx i + Iωy j + Iωz k. The components of H may also be expressed by Eqs. 21–10, where the inertia tensor is assumed to be known. Equate the i, j, and k components of both expressions for H and consider ωx, ωy, and ωz to be unknown. The solution of these three equations is obtained provided the determinant of the coefficients is zero. Show that this determinant, when expanded, yields the cubic equation
I3 – (Ixx + Iyy + Izz)I2
Want to see the full answer?
Check out a sample textbook solutionChapter 21 Solutions
Engineering Mechanics: Dynamics (14th Edition)
- An aeroplane makes a complete half circle of 50 metres radius, towards left, when flying at 200 km per hour. The rotary engine and the propeller of the plane has a mass of 400 kg with a radius of gyration of 300 mm. The engine runs at 2400 r.p.m. clockwise, when viewed from the rear. Find the gyroscopic couple on the aircraft and state its effect on it. What will be the effect, if the aeroplane turns to its right instead of to the left ?arrow_forwardA cylinder with rotational inertia 0.050 kg m2 is spinning with angular velocity 25 rad/s, clockwise with negligible friction about a vertical axis. A second cylinder that is not spinning is centered above and dropped onto the spinning cylinder. Kinetic friction produces torque of amount 0.60 Nm that affects both objects until the two cylinders spin together at angular velocity 18 rad/s, clockwise. (a) Determine the amount of time it takes for friction to cause the two objects to spin together.(b) Determine the rotational inertia of the second cylinder that was dropped onto the first.(c) Determine the total amount of heat and sound energy resulting from the interaction of the cylinders.arrow_forwardAn object is formed by attaching a uniform, thin rod with a mass of mr = 6.86 kg and length L = 5.76 m to a uniform sphere with mass ms = 34.3 kg and radius R = 1.44 m. Note ms = 5mr and L = 4R. 1) What is the moment of inertia of the object about an axis at the left end of the rod? kg-m2 2) If the object is fixed at the left end of the rod, what is the angular acceleration if a force F = 463 N is exerted perpendicular to the rod at the center of the rod? rad/s2arrow_forward
- In the helicopter shown; a vertical tail propeller is used to pre- vent rotation of the cab as the speed of the main blades is changed. Assuming that the tail propeller is not operating determine the final angular velocity of the cab after the speed of the main blades has been changed from I80 to 240 rpm. (The speed of the main blades is measured relative to the cab, and the cab has a centroidal moment of inertia of 650 lb.ft.s2. Each of the four main blades is assumed to be a slender rod 14 ft weighing 55 lb.)arrow_forwardThe pendulum consists of the 4-kgkg slender rod and the 5.5-kgkg thin plate. Suppose that aaa = 1.5 mm , bbb = 0.5 mm , and hhh = 2.4 mm A: Determine the location of the center of mass G of the pendulum. B: Determine the mass moment of inertia of the pendulum about an axis perpendicular to the page and passing through G.arrow_forwardThe total mass of the Baja car and driver, including the wheels, is 250 kg. Each pair of 58-cm radius wheels and the axle has a total mass of 20 kg and a mass moment of inertia of 2.9 kg·m2 . The center of gravity of the driver and Baja body (not including the wheels) is located x = 0.70 m from the rear axle A and y = 0.55 m from the ground. The wheelbase is L = 1.60 m. If the engine exerts a torque of 500 N·m on the rear axle, what is the car’s acceleration?arrow_forward
- Given an arbitrary body and three rectangular axes x, y, and z,, prove that the mass moment of inertia of the body with respect to any one of the three axes cannot be larger than the sum of the mass moments of inertia of the body with respect to the other two axes. That is, prove that the inequality Ix ≤ Iy +Iz and the two similar inequalities are satisfied. Furthermore, prove that Iy _> 1/2Ix if the body is a homogeneous solid of revolution, where x is the axis of revolution and y is a transverse axis.arrow_forwardA turbine rotor is mounted on a stepped shaft that is fixed at both ends as shown in The torsional stiffnesses of the two segments of the shaft are given by ka = 3,000 N-m/rad and ka = 4.000 N-m/rad. The turbine generates a harmonic torque given by M(t) = M, cos at about the shaft axis with M, = 200 N-m and e = 500 rad/s. The mass moment of inertia of the rotor about the shaft axis is Jo = 0.05 kg-m. Assuming the equivalent torsional damping constant of the system as e, = 2.5 N-m-s/rad, determine the steady-state response of the rotor, )arrow_forwarda rotating shaft carries four unbalanced masses 18 kg, 14 kg, 16 kg and 12 kg at radii 50 mm, 60 mm, 70 mm and 60 mm respectively. the 2nd, 3rd and 4th masses revolve in planes 80 mm, 160 mm and 280 mm respectively measured from the plane of the first mass and are angularly located at 60°, 135° and 270° respectively measured clockwise from the first mass looking from this mass end of the shaft. the shaft is dynamically balanced by two masses, both located at 50 mm radii and revolving in planes mid-way between those of 1st and 2nd masses and midway between those of 3rd and 4th masses. determine, graphically or otherwise, the magnitudes of the masses and their respective angular positions.arrow_forward
- A thin square plate with dimensions of 0.65 m x 0.65 m and a mass of 5.00 kg is supported from pins at points A and B as shown in Figure 1.1 (Position 1).a. what's its moment of inertia about its center of mass?b. If the pin at point B is removed and the plate freely swings around pin A, what's the magnitude of theplate's angular velocity once it has rotated in a clockwise direction by 90° as shown in Figure 1.2 (Position 2)?c. what's the magnitude of the maximum angular velocity attained by the plate as it swings freely?(Hint: The maximum angular velocity will occur after rotating 45° clockwise upon release.)arrow_forwardA 50 N force is applied 50 cm from the axis of rotation of a rigid body. The body starts from rest and completes 5 revolutions in 2.5 s. Calculate the Moment of Inertia of the body.arrow_forwardA 7.5-kg disk A radius 0.6 m initially rotating clockwise at 300 rev/min is engaged with an 8.5-kg disk B radius 0.4 m initially rotating counter-clockwise at 700 rev/min, where the moment of inertia of a disk is given as I=1/2mr^2. Determine their combined angular speed (in rpm) and direction of rotation after the meshing of the two disk.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