What is the transfer-of-axis relation for the product of inertia I_xy about a set of x-y axes, given the product of inertia I_bar_xy about parallel centroidal axes and the transfer distances d_x and d_y?
Explanation
Similar to the parallel-axis theorem for moments of inertia, there is a corresponding theorem for products of inertia. The product of inertia about a general axis pair is the sum of the product of inertia about the parallel centroidal axis pair and a transfer term m*d_x*d_y.
Other questions
What physical property does the mass moment of inertia, I, represent for a rigid body?
What is the fundamental integral that defines the mass moment of inertia, I, of a body about a given axis?
What are the correct dimensions for the mass moment of inertia?
The radius of gyration, k, is a measure of the distribution of a body's mass about an axis. How is it defined in terms of the body's mass, m, and its moment of inertia, I?
What are the two necessary conditions for using the parallel-axis theorem, I = I_bar + md^2, to find the moment of inertia about an axis?
For a thin flat plate of uniform thickness lying in the x-y plane, what is the relationship between its mass moments of inertia (I_xx, I_yy, I_zz) about the coordinate axes?
When using the composite body method to find the moment of inertia, how should a part of the body that represents a hole or removed material be handled?
What is the mass moment of inertia of a homogeneous solid sphere of mass 'm' and radius 'r' about an axis passing through its center (a diameter)?
What is the mass moment of inertia of a homogeneous right-circular cylinder of mass 'm' and radius 'r' about its central longitudinal axis?
What is the key difference between moments of inertia and products of inertia regarding their possible sign?
How is the product of inertia I_xy for a rigid body defined?
What is meant by the term 'principal axes of inertia' for a rigid body at a given origin?
What is the moment of inertia I_xx of a homogeneous rectangular parallelepiped of mass m and dimensions a, b, and l (along the y, z, and x axes respectively) about its centroidal x0-axis?
A slender rod of mass 'm' and length 'L' has a moment of inertia about its centroidal axis of (1/12)mL^2. What is its moment of inertia about a parallel axis located at one end of the rod?
For a thin homogeneous plate, the mass moment of inertia I_zz about the z-axis (perpendicular to the plate) can be related to the polar moment of inertia I_z of the plate's area. What is this relationship if the plate has density 'rho' and thickness 't'?
What does the inertia tensor, or inertia matrix, represent for a rigid body?
If a body has an axis of symmetry, what can be concluded about its products of inertia?
What is the radius of gyration 'k' for a homogeneous right-circular cylinder of radius 'r' about its central axis?
In Sample Problem B/5, a solid of revolution is generated where the radius y is proportional to x^2 (y = (r/h^2) * x^2). What is its mass moment of inertia I_xx about the axis of revolution?
What is the moment of inertia of a thin spherical shell of mass 'm' and radius 'r' about any diameter?
The parallel-axis theorem for the radius of gyration is given by k^2 = k_bar^2 + d^2. What does k_bar represent in this equation?
Consider a uniform thin plate with an xy-plane of symmetry. Which products of inertia must be zero?
What is the moment of inertia I_yy of a uniform thin triangular plate of mass m, height h, and base b, about the y-axis? Base your answer on the information for a thin plate of this shape in the appendix tables.
What is the moment of inertia I_xx for a uniform thin triangular plate of mass m and height h about its base (the x-axis)?
What is the moment of inertia of a uniform slender rod of mass 'm' and length 'L' about its centroidal axis, perpendicular to the rod?
A complete ring (torus) of mass m is generated by revolving a circular area of radius 'a' about an axis, where the distance from the axis to the center of the circular area is 'R'. What is its moment of inertia about the generating axis (the axis of revolution)?
In the context of three-dimensional motion, why is a double subscript notation (e.g., I_xx, I_yy) used for moments of inertia instead of a single subscript (e.g., I_O)?
For a homogeneous half-cylindrical shell of mass 'm' and radius 'r', what is its moment of inertia I_zz about the central longitudinal z-axis?
What are the three roots (I1, I2, I3) obtained by solving the determinant equation derived from the inertia matrix?
A homogeneous solid of mass m is formed by revolving the 45-degree right triangle shown in Problem B/20 about the z-axis. What is its radius of gyration about the z-axis?
What is the product of inertia I_xy for the uniform slender rod of mass m and length L shown in Problem B/64, which lies in the xy-plane and makes an angle theta with the x-axis?
A half-ring of mass m has its diametral axis labeled a-a. According to Problem B/39, what is its moment of inertia about this axis a-a?
Consider a thin conical shell of mass m and radius r. What is its moment of inertia about the z-axis (axis of symmetry)?
What is the moment of inertia of a solid right-circular cone of mass m and base radius r about its central axis?
The homogeneous solid of revolution in Problem B/13 is defined by y = kx^(1.5). What is its mass moment of inertia about the x-axis (axis of revolution)?
For the S-shaped piece in Problem B/67, formed from a slender rod, what would be the value of the product of inertia I_xy?
The moment of inertia of a solid homogeneous cylinder of radius r about a parallel axis at a distance d is approximated by md^2. According to Problem B/33, what percentage error 'e' results if d = 2r?
For the homogeneous thin plate in Sample Problem B/4, with a parabolic edge y = (h/b^(1/2)) * x^(1/2), what is the mass moment of inertia I_xx about the x-axis?
What is the primary method used in Sample Problem B/6 to find the products of inertia for the bent plate?
A homogeneous solid semi-ellipsoid of revolution of mass m is generated by revolving a semi-ellipse about the x-axis. As in Problem B/19, what is its moment of inertia about the x-axis?
What is the moment of inertia of a solid steel half-cylinder about its longitudinal axis x-x as shown in problem B/47?
For the steel half-cylinder in Problem B/44, which product of inertia is non-zero for the given y-z plane of symmetry?
What is the moment of inertia of a solid cube of mass m and side length 'a' about a diagonal axis A-A passing through opposite corners, as in Problem B/70?
The clock pendulum in Problem B/48 consists of a slender rod of length l and mass m, and a bob of mass 7m. Neglecting the bob's radius, what is the moment of inertia I_O about the pivot O in terms of the bob's position x?
For the assembly of three small spheres in Problem B/57, each of mass m, what is the product of inertia I_xy?
What is the moment of inertia for a quarter-circular rod of mass m and radius r about the x-axis, as shown in Table D/4?
In Problem B/77, a thin plate is formed into an L-shape with legs of length b and 2b. The plate has a mass per unit area of rho. What is the principal moment of inertia I1 (maximum value)?
The mass moment of inertia of a composite body is the sum of the moments of inertia of its individual parts about what axis?
For the half-cylindrical shell of mass m and radius r in problem B/21, what is the moment of inertia about the axis a-a (a diameter of the base)?