Theorems of Parallel and Perpendicular Axis: The theorems of parallel and perpendicular axis are two important theorems in physics that relate to the moments of inertia of rigid bodies.

Theorems of Parallel and Perpendicular Axis

The Parallel Axis Theorem

The parallel axis theorem states that the moment of inertia of a rigid body about any axis parallel to its center of mass axis is equal to the moment of inertia about the center of mass axis plus the product of the mass of the body and the square of the distance between the two axes. Mathematically, this can be expressed as:

I = Icm + md²

Where,

• I is the moment of inertia about the parallel axis
• Icm is the moment of inertia about the center of mass axis
• m is the mass of the body
• d is the distance between the two axes.

This theorem is useful in determining the moment of inertia of irregularly shaped objects, as it allows us to calculate the moment of inertia about a parallel axis that is easier to calculate.

The Perpendicular Axis Theorem

The perpendicular axis theorem states that the moment of inertia of a rigid body about any axis perpendicular to its plane of symmetry is equal to the sum of the moments of inertia about two perpendicular axes lying in the plane of symmetry. Mathematically, this can be expressed as:

Iz = Ix + Iy

Where,

• Iz is the moment of inertia about the axis perpendicular to the plane of symmetry
• Ix and Iy are the moments of inertia about the two perpendicular axes lying in the plane of symmetry

This theorem is useful in determining the moment of inertia of planar objects, as it allows us to calculate the moment of inertia about an axis perpendicular to the plane of symmetry, which is often easier to calculate than the moments of inertia about the other two axes lying in the plane of symmetry.

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By Team Learning Mantras