The general expression for the angular momentum H of a rigid body rotating with angular velocity ω involves an array of moments and products of inertia known as the inertia matrix. If the coordinate axes are the principal axes of inertia, how is the angular momentum H expressed?
Explanation
This question tests the simplified form of the angular momentum vector when the analysis is performed using the principal axes of inertia, where the off-diagonal terms (products of inertia) of the inertia matrix are zero.
Other questions
What is the primary distinction between finite rotations and infinitesimal rotations of a rigid body when treated as vectors?
For a rigid body rotating about a fixed point, the angular acceleration α is the time derivative of its angular velocity ω (α = dω/dt). In three-dimensional motion, what does the vector α represent?
An arm OA of length 0.8 m is pivoted about the horizontal x-axis, and the entire assembly rotates about the vertical z-axis. The arm is being raised at a constant rate of 4 rad/s. The assembly rotates about the z-axis at a constant speed of 60 rev/min (which is 6.28 rad/s). What is the angular acceleration α of the arm OA?
In the context of a rigid body's inertial properties, what are the axes for which the products of inertia (Ixy, Ixz, Iyz) vanish?
What is the phenomenon called when the axis of a spinning rotor itself rotates about another axis under the influence of an external couple?
For the steady precession of a symmetrical gyro, the simplified relationship between the applied couple M, the spin velocity p, the precessional velocity Ω, and the moment of inertia about the spin axis I is M = I*p*Ω. What is the relationship between the vectors M, p, and Ω?
A 1000 kg turbine rotor in a ship has a radius of gyration of 200 mm and spins at 5000 rev/min counterclockwise when viewed from the stern. The ship makes a left turn of 400 m radius at a speed of 25 knots (12.85 m/s). What is the gyroscopic effect on the ship?
What are Euler's equations in the context of three-dimensional rigid-body dynamics?
What type of motion is described as parallel-plane motion?
For a rigid body in general motion, the velocity of a point A can be expressed relative to another point B on the same body using the equation vA = vB + vA/B. What does the term vA/B represent?
The general expression for the kinetic energy T of a rigid body is T = (1/2)m*v_bar^2 + Σ(1/2)m_i*rho_i_dot^2. For a rigid body, what does the second term, representing motion relative to the mass center, become?
A symmetrical rotor experiences torque-free motion in space (ΣMG = 0). It spins with angular velocity ω and precesses about the constant angular momentum vector HG. This motion is described as steady precession. What determines if the precession is direct (body cone rolls on the outside of the space cone)?
A solid circular cylinder of mass m and radius r rotates about its central axis O-O. What is its mass moment of inertia I about this axis?
The parallel-axis theorem for mass moments of inertia is I = I_bar + md^2. What is the necessary condition for this theorem to be valid?
A bent plate consists of two rectangular parts. Part A is 0.100 m by 0.125 m. Part B is 0.075 m by 0.150 m. The plate has a mass of 70 kg per square meter and revolves about the z-axis at 30 rad/s. What is the component of its angular momentum H about the x-axis?
What is the key advantage of using rotating reference axes for kinematic analysis over translating reference axes?
The moment equation for three-dimensional motion ΣM = dH/dt requires the time derivative of the angular momentum vector H. If H is expressed in terms of components in a rotating frame x-y-z with angular velocity Ω, what is the correct expression for ΣM?
A motor housing and its bracket rotate about the vertical Z-axis at a constant rate Ω = 3 rad/s. The motor shaft inside has a constant spin p = 8 rad/s about its own axis (the y-axis) relative to the housing. At an instant when the housing's y-axis is aligned with the global Y-axis, what is the angular velocity ω of the motor disk?
For the same motor system as in the previous question (Ω = 3k rad/s, p = 8j rad/s), what is the angular acceleration α of the motor disk, assuming steady precession?
When a rigid body is spinning about an axis of symmetry, it is often convenient to choose one axis of the reference system to coincide with the spin axis and allow the other two axes not to turn with the body. What must be accounted for in the angular momentum calculation in this case?
What does the 'transfer principle for angular momentum', HP = HG + r × G, allow one to do?
Euler's equation for the moment about the x-principal axis is ΣMx = Ixx*ω_dot_x - (Iyy - Izz)*ω_y*ω_z. What do the terms (Iyy - Izz)*ω_y*ω_z represent?
A proposed space station, approximated by four uniform spherical shells, is designed to rotate about its z-axis at one revolution every 4 seconds. The moment of inertia about the spin axis is Izz = (56/3)mr^2 and the transverse moment of inertia is Ixx = (32/3)mr^2. If the axis of rotation deviates slightly from a fixed orientation, what is the ratio of the precession frequency (wobble) to the spin frequency?
How is the moment of inertia of a thin flat plate about an axis normal to the plate (Izz) related to its moments of inertia about axes lying in the plane of the plate (Ixx and Iyy)?
What is the mass moment of inertia of a slender rod of mass m and length l about an axis through its center and perpendicular to the rod?
A homogeneous solid sphere of mass m and radius r has a moment of inertia I about a diameter. What is the value of I?
In the context of three-dimensional dynamics, which statement correctly describes the products of inertia (e.g., Ixy)?
A mechanism consists of two cranks, CB and DA, connected by a link AB. Crank CB rotates with angular velocity ω1 = 6i rad/s. At a certain instant, the velocity of point B is vB = 600i mm/s. The velocity of point A is vA = 50*ω2 j mm/s. The relative velocity is given by vA = vB + ω_n × rA/B, where rA/B = (-50i + 100j + 100k) mm. From this, the kinematic relations are -6 = ω_ny + ω_nz and ω2 = 6 rad/s. What additional constraint is required to solve for the angular velocity of link AB, ω_n?
For a thin homogeneous plate of mass m, the mass moments of inertia about axes x and y lying in the plate are Ixx and Iyy. What is the mass moment of inertia Izz about the z-axis which is normal to the plate?
In torque-free motion of a symmetric body, if the moment of inertia about the spin axis (I) is greater than the transverse moment of inertia (I0), the precession is described as direct. What does this mean for the body and space cones?
A slender rod is bent into a shape with segments of length b parallel to the x, y, and z axes. The rod has a mass per unit length ρ and rotates about the z-axis with an angular velocity ω. What is the angular momentum HO about the origin O?
A top is spinning with spin velocity p about its axis, which is inclined at an angle θ to the vertical. It is also precessing with velocity Ω about the vertical axis. The moment about the pivot O is due to its weight mg. The simplified gyroscopic equation is mgr = I*p*Ω. What can be concluded about the precession velocity Ω?
What is the primary difference between the equations of motion for parallel-plane motion and general plane motion (as in Chapter 6)?
A solid half-circular cylinder of mass m revolves about the z-axis with an angular velocity ω. How would you calculate its angular momentum H about the origin O?
When is it permissible to use the simplified gyroscopic relation M = I p Ω?
A car with rear-wheel drive makes a sharp right turn. The engine's crankshaft spins clockwise when viewed from the front. What is the primary gyroscopic effect of the crankshaft on the car's stability?
When a body rotates about a fixed axis, its velocity v is given by v = ω × r. What is the corresponding expression for its acceleration a?
A homogeneous rectangular block of mass m has dimensions a, b, and length l. What is its moment of inertia I_x0x0 about the centroidal x0-axis?
What is the primary characteristic of the motion of a body undergoing translation in three dimensions?
A flight simulator is mounted on hydraulic actuators. For a particular instant, its center B has a velocity of 3.2 ft/sec and acceleration of 4 ft/sec^2, both in the y-direction. Simultaneously, the simulator has angular velocities ωx = 1.4 rad/sec and ωy = 1.2 rad/sec. What is the magnitude of the velocity of point A, located at (-60i) inches from B?
What does a non-zero product of inertia, such as Ixz, physically represent in the context of parallel-plane motion (rotation about the z-axis)?
An electric motor and disk are rotating about a vertical Z-axis at a constant rate Ω = 2π rad/sec (60 rev/min). The motor also spins about its own local z-axis (tilted at 30 degrees to Z) at ω0 = 4π rad/sec. What is the magnitude of the angular acceleration α of the disk?
A spool A rotates about its axis with an angular velocity of 20 rad/s. Simultaneously, the assembly rotates about the vertical axis with an angular velocity of 10 rad/s. The spool's axis is tilted at 60 degrees from the vertical. What is the magnitude of the total angular velocity of the spool?
What is the inertia tensor or inertia matrix?
The kinetic energy of a body pivoted about a fixed point O can be expressed as T = (1/2)ω ⋅ HO. What does HO represent in this equation?
A solid right-circular cone of mass m, height h, and base radius r rolls on a flat surface without slipping. The center of its base moves in a circle with constant speed v. What physical principle determines the relationship between the cone's angular velocity ω and the speed v?
An aircraft landing gear wheel is spinning at a rate corresponding to a takeoff speed of 200 km/h (55.6 m/s). The wheel has a radius of 0.46 m (920mm diameter). During retraction, the gear assembly rotates at a rate of 30 degrees per second (π/6 rad/s). What is the magnitude of the gyroscopic moment on the wheel axle?
What is the primary purpose of using a gimbal mounting for a gyroscope in an inertial guidance system?
The general moment equation ΣM = dH/dt is valid for which of the following reference points for ΣM and H?