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In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia 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 , , and (which meet at origin ) so that the body lies in the plane, and the 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[1]
This rule can be applied with the parallel axis theorem and the stretch rule to find moments of inertia for a variety of shapes.
If a planar object (or prism, by the stretch rule) has rotational symmetry such that and are equal, then the perpendicular axes theorem provides the useful relationship:
DerivationWorking in Cartesian co-ordinates, the moment of inertia of the planar body about the axis is given by[2]:On the plane, , so these two terms are the moments of inertia about the and axes respectively, giving the perpendicular axis theorem.
What else can I help you with?
How can one determine the moments of inertia for a given object, and what methods are used in deriving moments of inertia?
To determine the moments of inertia for an object, one can use mathematical formulas or physical experiments. The moment of inertia depends on the shape and mass distribution of the object. Common methods for deriving moments of inertia include integration, parallel-axis theorem, and the perpendicular-axis theorem. These methods involve calculating the distribution of mass around an axis to determine how the object resists rotational motion.
What is the significance of the intermediate axis theorem in the study of rotational motion and stability?
The intermediate axis theorem is important in the study of rotational motion and stability because it explains the behavior of an object rotating around its intermediate axis. This theorem helps predict how objects will rotate and maintain stability, especially in situations where the rotation is not around the principal axes. Understanding this theorem is crucial for analyzing the motion and stability of rotating objects in various scenarios.
If moment of inertia of a body change of axis of rotation?
If the moment of inertia of a body changes due to a change of axis of rotation, the new moment of inertia can be calculated using the parallel axis theorem. This theorem states that the moment of inertia about a new axis parallel to the original axis can be found by adding the mass of the body multiplied by the square of the distance between the two axes.
How does the object move when it is rotated along an axis that is perpendicular to the page, transitioning it from being in the page to out of the page?
When an object is rotated along an axis that is perpendicular to the page, it moves from being flat on the page to sticking out of the page.
What is the direction of the precession of the gyroscope?
The direction of precession of a gyroscope is perpendicular to the axis of rotation.
State and prove perpendicular axis and parallel axis theorem?
In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia 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 , , and (which meet at origin ) so that the body lies in the plane, and the 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[1]This rule can be applied with the parallel axis theorem and the stretch rule to find moments of inertia for a variety of shapes.If a planar object (or prism, by the stretch rule) has rotational symmetry such that and are equal, then the perpendicular axes theorem provides the useful relationship:
State and prove the parallel axis theorem?
the moment of inertia of a body about a given axis is equal to the sum of its moment of inertia about a parallel axis passing through its centre of mass and the product of its mass and square of perpendicular distance between two axis Iz=Ix+Iy
State Perpendicular axis theorem?
if xx and yy be the two axes and the moment of inertia of them be Ixx and Iyy then the moment of inertia about the zz axes will be Izz
Which axis of a hyperbola is perpendicular to the conjugate axis?
The transverse axis is perpendicular to the conjugate axis.
Which plane is Perpendicular to the longitudinal axis?
The transverse plane is perpendicular to the longitudinal axis.
Which axis of a hyperbola is perpendicular to the transverse axis?
the conjugate axis
Can a vertical line be perpendicular to the x-axis?
Yes because the y axis is perpendicular to the x axis at the origin which is (0, 0)
What is the perpendicular bisector of transverse axis?
It is the conjugate axis or the minor axis.
When a point lies on the y-axis what do you know about it's x-coordinate .?
The x-coordinate of any point on the y-axis is 0. The y-axis is a line perpendicular to the x-axis. Any point on a line perpendicular to the x-axis has the same x-coordinate. The y-axis is the line perpendicular to the x-axis through 0, and has the equation x = 0; similarly, the x-axis is the line perpendicular to the y-axis through 0 and has the equation y = 0.
What is the moment of inertia of a rod?
The moment of inertia of a cube depends on what its axis of rotation is. About an axis perpendicular to one of its sides and through the centre of the cube is (ML2)/6. Where M is the Mass of the Cube and L the length of its side. Due to the symmetry of the cube, you can find the Moment of Inertia about almost any other axis by using Parallel and Perpendicular Axis Theorems.
Is y equals -2 a parallel line or a perpendicular line?
y=-2 is parallel to the x-axis and perpendicular to the y-axis.
What is the distance covered by the earth on its axis when it rotates?
Its own circumference perpendicular to its axis
