Angular Momentum


Angular momentum and its properties were devised over time by many of the great minds in physics. Newton and Kepler were probably the two biggest factors in the evolution of angular momentum. Angular momentum is the force which a moving body, following a curved path, has because of its mass and motion. Angular momentum is possessed by rotating objects. Understanding torque is the first step to understanding angular momentum.

Torque is the angular "version" of force. The units for torque are in Newton-meters. Torque is observed when a force is exerted on a rigid object pivoted about an axis and. This results in the object rotating around that axis. "The torque ? due to a force F about an origin is an inertial frame defined to be ? ? r x F"1 where r is the vector position of the affected object and F is the force applied to the object.

To understand angular momentum easier it is wise to compare it to the less complex linear momentum because they are similar in many ways. "Linear momentum is the product of an object's mass and its instantaneous velocity. The angular momentum of a rotating object is given by the product of its angular velocity and its moment of inertia. Just as a moving object's inertial mass is a measure of its resistance to linear acceleration, a rotating object's moment of inertia is a measure of its resistance to angular acceleration."2 Factors which effect a rotating object's moment of inertia are its mass and on the distribution of the objects mass about the axis of rotation. A small object with a mass concentrated very close to its axis of rotation will have a small moment of inertia and it will be fairly easy to spin it with a certain angular velocity. However if an object of equal mass, with its mass more spread out from the axis of rotation, will have a greater moment of inertia and will be harder to accelerate to the same angular velocity.3

To calculate the moment of inertia of an object one can imagine that the object is divided into many small volume elements, each of mass ?m. "Using the definition (which is taken from a formula in rotational energy) I=?ri2?mi and take the sum as ?m?0 (where I is the moment of inertia and ri is the perpendicular distance of the infinitely small mass' distance from the axis of rotation). In this limit the sum becomes an integral over the whole object:
I = lim ?ri2?mi = ? r2 dm. To evaluate the moment of inertia using this equation it is necessary to express each volume element (of mass dm) in terms of its coordinates. It is common to define a mass density in various forms. For a three-dimensional object, it is appropriate to use the volume density, that is, mass per unit of volume:
? = lim ?m = dm
?v?0 ?d dV
dm = ? dV therefore: I = ? ?r2 dV."5

Since every different shape has all of its mass in different places relative to the axis of rotation a different final, simplified formula results for every shape. The shapes that will be focused on (in presentation) are the: hoop of a cylindrical shell, solid cylinder or disk, and the rectangular plane, with formulas: ICM = MR2, ICM = 1/2 MR2 , and ICM = 1/12 M(a2 + b2) respectively (see diagrams on sheet titled "Moments of Inertia of Some Rigid Objects").

Similar to the Law of Conservation of Linear Momentum is the Law of Conservation of Angular Momentum. This law applies to rotating systems that have no external torques or moments applied to them. This law helps to explain why a rotating object will start to spin faster (with a greater angular velocity) if all or some of its mass is brought inward towards its rotating axis or why it would start to rotate with a decreased angular velocity if some of its mass is "spread" out away from its rotating axis. An example of this is the slowly spinning figure skater who pulls his arms close to himself and suddenly speeds up his angular velocity. When he wants to decrease his angular velocity (or his velocity of rims) he merely spreads out his arms again and just as suddenly as he sped up, he can slow down. If the mass of the skaters hands is known as well