1.2 Basic Formula and Units

The moment of inertia (I) is calculated using the formula I = ∫ r² dm, where r is the perpendicular distance from the axis of rotation, and dm is the differential mass element. This integral is evaluated over the entire mass of the object. For uniform shapes, density (ρ) and volume (V) simplify the calculation, resulting in formulas specific to each geometry. The unit of moment of inertia is kg·m² in the International System (SI), reflecting mass and distance squared. In U.S. Customary Units, it is expressed as slug·ft². Understanding the basic formula is foundational for applying pre-derived formulas to common shapes like rods, plates, and spheres, ensuring accurate calculations in engineering and physics applications. This equation underscores the importance of mass distribution in rotational dynamics, emphasizing why different shapes have distinct moment of inertia values.

Moment of Inertia for Common Shapes

This section provides formulas for calculating the moment of inertia of common geometric shapes, such as solid spheres, thin rods, rectangular plates, circular rings, and hollow cylinders, essential for engineering and physics applications.

2.1 Solid Sphere

The moment of inertia of a uniform solid sphere rotating about its central axis is given by the formula ( I = rac{2}{5} MR^2 ), where ( M ) is the mass and ( R ) is the radius. This result is derived by integrating the differential moment of inertia over the entire volume of the sphere, taking into account the perpendicular distance of each mass element from the axis of rotation. The formula assumes a uniform mass density and applies to symmetrical axes passing through the center of the sphere. For axes located at a distance ( d ) from the center, the parallel axis theorem can be used to adjust the moment of inertia. This fundamental formula is widely used in engineering and physics to analyze rotational motion and stress in spherical objects like gears and flywheels.

2.2 Thin Rod

The moment of inertia of a thin rod rotating about an axis perpendicular to its length and passing through its center is given by ( I = rac{1}{12} ML^2 ), where ( M ) is the mass and ( L ) is the length of the rod. For rotation about an axis passing through one end and perpendicular to the rod, the formula becomes ( I = rac{1}{3} ML^2 ). These formulas are derived by integrating the contributions of infinitesimal mass elements along the rod’s length. The thin rod approximation assumes the rod’s thickness is negligible compared to its length, simplifying the calculations. These expressions are fundamental in analyzing rotational motion in systems like balances, pendulums, and structural beams.

2.3 Rectangular Plate

The moment of inertia of a rectangular plate varies depending on the axis of rotation. For a thin rectangular plate of mass ( M ), length ( a ), and width ( b ), the moment of inertia about an axis perpendicular to the plate and passing through its center is ( I = rac{1}{12} M (a^2 + b^2) ). If the axis lies along one of the edges, the formula becomes ( I = rac{1}{3} M b^2 ) for rotation about the length ( a ). These formulas are derived by integrating the contributions of infinitesimal mass elements across the plate’s area. The rectangular plate’s moment of inertia is essential in structural and mechanical analyses, particularly for objects like flat gears or plates in bridges, where rotational stiffness is critical. The parallel axis theorem can also be applied to calculate the moment of inertia about any other axis.

2.4 Circular Ring

The moment of inertia of a circular ring, or hoop, is a fundamental concept in rotational dynamics. For a thin circular ring of mass ( M ) and radius ( R ), the moment of inertia about an axis passing through its center and perpendicular to the plane of the ring is given by ( I = MR^2 ). This formula arises because the entire mass is distributed at a constant distance ( R ) from the axis of rotation. The derivation involves integrating the contributions of infinitesimal mass elements around the ring, each at a distance ( R ) from the center. This result is widely used in engineering and physics, particularly in systems involving rolling without slipping or rotational kinetic energy. The circular ring’s moment of inertia is also a building block for analyzing more complex shapes and structures.

2.5 Solid Cylinder

The moment of inertia of a solid cylinder is a critical parameter in rotational dynamics. For a solid cylinder of mass ( M ) and radius ( R ), the moment of inertia about its central axis (perpendicular to the axis of rotation) is ( I = rac{1}{2}MR^2 ). This result is derived from integrating the contributions of infinitesimal mass elements, each at a distance ( r ) from the axis, over the entire volume of the cylinder. The factor ( rac{1}{2} ) arises due to the symmetry of the mass distribution. For rotation about an axis perpendicular to the cylinder’s length and passing through its center, the moment of inertia remains the same. This formula is widely applied in engineering and physics, particularly in systems like flywheels, engines, and turbines, where rotational energy and stability are crucial. The solid cylinder’s moment of inertia serves as a foundational result for analyzing more complex rotational systems.

2.6 Triangular Plate

The moment of inertia of a triangular plate varies depending on the axis of rotation. For a triangular plate of mass ( M ) and base length ( L ), the moment of inertia about an axis perpendicular to the plane of the triangle and passing through its centroid is given by ( I = rac{1}{18}ML^2 ). This result is derived by integrating the contributions of infinitesimal mass elements across the plate’s area. The formula assumes a uniform mass distribution and a right triangle, though it can be extended to other triangular shapes with appropriate adjustments. The moment of inertia increases significantly when the axis of rotation is moved away from the centroid, following the parallel axis theorem. This property is essential in structural analysis and mechanical design, where understanding rotational dynamics is critical. The triangular plate’s moment of inertia is a fundamental concept in calculating stresses and deflections in engineering systems.

2.7 Hoop

A hoop is a thin, circular ring with its mass distributed uniformly around its circumference. The moment of inertia of a hoop rotating about an axis perpendicular to its plane and passing through its center is given by ( I = MR^2 ), where ( M ) is the total mass and ( R ) is the radius. This formula arises because all the mass elements are at a constant distance ( R ) from the axis of rotation. For a hoop, the moment of inertia is maximized compared to other shapes of similar mass and radius due to the mass being located far from the axis. This property makes hoops ideal for applications requiring high rotational inertia, such as gyroscopes. The formula is derived using integration, considering small mass elements ( dm ) at a distance ( R ) from the axis. The hoop’s moment of inertia is a fundamental concept in rotational dynamics and engineering design.