This equation is exactly Equation 10.25 but with the torque and angular acceleration as vectors. Theorem of parallel axes is applicable for any type of rigid body whether it is a two dimensional or three dimensional, while the theorem of perpendicular is applicable for laminar type or two I dimensional bodies only. Identifying the first term on the left as the sum of the torques, and m r 2 as the moment of inertia, we arrive at Newton’s second law of rotation in vector form: net I. horizontal strips when you want to find the moment of inertia about the x axis and vertical strips for the moment of inertia about the y axis. Magnetic Torque: B NBextIA sin(), angle between area vector. Where I x and I y are the moments of inertia of plane lamina about perpendicular axes X and Y respectively which lie in the plane lamina an intersect each other. The most straightforward approach is to use the definitions of the moment of inertia (10.1.3) along with strips parallel to the designated axis, i.e. Area vector: A is normal to plane of loop, typically same direction as magnetic moment. The moment of inertia of any two dimensional body about an axis perpendicular to its plane (I z) is equal to the sum of moments of inertia of the body about two mutually perpendicular axes lying in its own plane and intersectingĮach other at a point, where the perpendicular axis passes through it.

Where I is the moment of inertia about the arbitrary axis, I cM is moment of inertia about the parallel axis through the centre of mass, M is the total mass of the object and r is the perpendicular distance between the axis. 10.4: Moment of Inertia of Composite Shapes. 10.2: Moments of Inertia of Common Shapes. The moment of inertia of any object about any arbitrary axes is equal to the sum of moment of inertia about a parallel axis passing through the centre of mass and the product of mass of the body and the square of the perpendicular distance between They have different units however, and the intended moment of inertia can easily be determined from context or with a unit analysis. Therefore, moment of inertia I = MK 2 ⇒ K = √1/M The product of the mass of the body (M) and square of its radius gyration (K) gives the same moment of inertia of the body about rotational axis. The root mean square distance of its constituent particles from the axis of rotation is called the radius of gyration of a body.

Moment of inertia formula sheet