moment of inertia of a circle derivation

In doing so the force will cause the mass to rotate about O.Since tangential acceleration is related to angularĪs the turning effect is produced by torque τ, it would, therefore, be better to write the equation for rotation in terms of torque. A force F is acting on the mass perpendicular to the rod and hence this will accelerate the mass according to: Let us assume that the bearing at the pivot point O is frictionless.

Moment of inertia equationĬonsider a mass m attached to the end of a massless rod. Hence the kinetic energy of a body rotating about a fixed axis with angular velocity ω is ½ω², which corresponds to ½mv² for the kinetic energy of a body of mass m translated with velocity v.See also Routh’s rule the theorem of parallel axes.

The moment of inertia plays a role analogous to that of the inertial mass in the case of rectilinear and uniform motion. It is the scalar value of the longitudinal angular momentum of a rigid solid.įor a rigid body moving about a fixed axis, the laws of motion have the same form as those of rectilinear motion, with the moment of inertia replacing mass, angular replacing linear velocity, angular momentum replacing linear momentum, etc.

The moment of inertia reflects the mass distribution of a body or a system of rotating particles, with respect to an axis of rotation. The moment of inertia only depends on the geometry of the body and the position of the axis of rotation, but it does not depend on the forces involved in the movement.

Moment of inertia of a circle derivation