The kinetic energy of an object moving from one point to another is found by using the equation 1/2mv2 . To determine the kinetic energy of an object rotating about an axis, the object is broken down into small-mass objects that are treated as point masses. The sum of the contributions of each small-mass object provides the kinetic energy of the rotating object. As this is done, a new quantity called the moment of inertia is introduced, which acts similar to mass.
First, consider a point mass m that is orbiting in a circular path of radius r around some fixed point. Its kinetic energy is given by 1/2mv2 , where v is the magnitude of its velocity vector. Because the point mass repeats its motion each time it makes one complete orbit, its energy can be expressed in terms of an angular frequency, (omega) = v/r. The expression for the point mass's kinetic energy is then given by 1/2mr2 (omega)2 . In this expression, the quantity mr2 is called the moment of inertia (I) of the point mass. Generally, for an extended object, the kinetic energy can be written as 1/2I(omega)2 . The moment of inertia is the quantity that multiplies 1/2(omega)2 in the expression for kinetic energy.
In order to find an expression for the moment of inertia of an extended object, the object is broken down into small point masses, the moment of inertia is calculated for each point mass, and then added to find the total moment of inertia. For simple objects with constant density, this is an elementary math problem with simple formulas for the moment of inertia. For more complicated objects, such as Earth, the operation would have to be done using a computer. The main principle involved is that objects with greater mass farther away from the rotation axis have higher moments of inertia. For example, a hulahoop has a higher moment of inertia than a plastic disk of the same mass and radius, because all of the hulahoop's mass is all concentrated at its outer radius.
The moment of inertia acts much the way a mass does in linear motion. Essentially, mass opposes force, because for a constant force, heavier masses accelerate less. Similarly, for a constant torque, the rotational analogue of a force, objects with higher moments of inertia will have lower angular accelerations than objects of lower moments of inertia. In fact, the rotational kinematic equations are exact analogues of the linear kinematic equations; angle replaces position, angular frequency replaces velocity, angular acceleration replaces acceleration, torque replaces force, and angular momentum replaces momentum.
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