The Perpendicular Axis Theorem The Perpendicular axis theorem states that the centroid of a rigid body lies on a line perpendicular to the line of action...
The Perpendicular axis theorem states that the centroid of a rigid body lies on a line perpendicular to the line of action of a force applied to the body. This means that the total angular displacement of the body about the centroid will be equal to the angle of rotation produced by the applied force.
Example: A force applied perpendicular to the axis of a wheel will cause the wheel to rotate about the axle through the centroid of the wheel.
The perpendicular axis theorem can be proven using various geometric arguments. One approach is to consider a small segment of the body and analyze the forces acting on it. Another approach is to use the principle of moments of inertia to show that the angular displacement about the centroid is equal to the angle of rotation.
The perpendicular axis theorem has a wide range of applications in engineering mechanics. It is used to analyze the behavior of rigid bodies subjected to external forces, determine the natural frequency of a system, and design mechanisms for various machines.
Examples:
A car driving down a curved road follows a path that is perpendicular to the direction of the force applied to the car by the road.
A wheel spinning on a fixed axle is an example of a rigid body with its centroid located off the axis of rotation.
In a mechanical system, the perpendicular axis theorem can be used to determine the natural frequency of the system, which is the frequency at which the system oscillates back and forth