Question

What is perpendicular axis theorem.
Plzzz it is imp...​

Answer 1

Answer:

In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the second polar moment of area of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane. The axes must all pass through a single point in the plane.

Define perpendicular axes {\displaystyle x} , {\displaystyle y} , and {\displaystyle z} (which meet at origin {\displaystyle O} ) so that the body lies in the {\displaystyle xy} plane, and the {\displaystyle z} axis is perpendicular to the plane of the body. Let Ix, Iy and Iz be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that

{\displaystyle I_{z}=I_{x}+I_{y}}

This rule can be applied with the parallel axis theorem and the stretch rule to find polar moments of inertia for a variety of shapes.

If a planar object (or prism, by the stretch rule) has rotational symmetry such that {\displaystyle I_{x}} and {\displaystyle I_{y}} are equal, then the perpendicular axes theorem provides the useful relationship:

{\displaystyle I_{z}=2I_{x}=2I_{y}}

Working in Cartesian co-ordinates, the moment of inertia of the planar body about the {\displaystyle z} axis is given by:

{\displaystyle I_{z}=\int \left(x^{2}+y^{2}\right)\,dm=\int x^{2}\,dm+\int y^{2}\,dm=I_{y}+I_{x}}

On the plane, {\displaystyle z=0} , so these two terms are the moments of inertia about the {\displaystyle x} and {\displaystyle y} axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly.

Note that {\displaystyle \int x^{2}\,dm=I_{y}\neq I_{x}} because in {\displaystyle \int r^{2}\,dm} , r measures the distance from the axis of rotation, so for a y-axis rotation, deviation distance from the axis of rotation of a point is equal to its x co-ordinate.