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 as the distance from the skaters hands to his centre of mass and his angular velocity then the skaters angular momentum can be calculated using the formula: L= mvr + mvr (where L is the skaters angular momentum in Kg?m2/s, m is the mass of each hand in Kg, v is the velocity of rims, and r is the radius-arm length in this case). This formula can be derived from the linear equation of momentum which is: ? = mv (where ? is the momentum in Kg?m/s2, m is the mass in Kg, and v is the velocity in m/s2). L = mvr + mvr = 2mvr but v = r? (? is angular velocity in radians/second) ?L = 2mr2?
With the latter of the above equations all kinds of angular momentum problems can be solved. Since angular momentum is conserved in all cases that have zero outside forces the formula: L = L' becomes a very useful tool when trying to solve unknown variables. Angular momentum is conserved in many situations other than in figure skating. Gyroscopes, with no external forces, also show conservation of angular momentum.A gyroscope is any rotating body that exhibits two fundamental properties: gyroscopic inertia (rigidity in space) and precession (the tilting of the axis at ninety degree angles to any force inclined to alter the plane of rotation). These properties are present in all rotating bodies including planetary bodies like the earth, moon and sun. The term gyroscope is usually in reference to a spherical, wheel-shaped object that is universally mounted to be free to rotate in any direction. Gyroscopes are used to demonstrate the two properties of rotating bodies or to indicate movements in space. Gyroscopic Inertia can be explained using Newton's first law of motion, which states that a body tends to continue in its state of rest or uniform motion unless it is subject to some outside force. So the wheel of a gyroscope, once in motion, tends to rotate continuously in the same plane about the same axis in space like the Earth will continue to rotate around the sun, unless it is disturbed by an outside force or torque. Precession is observed when a force applied to a gyroscope changes the direction of the axis of rotation, the axis will move in a direction at right angles to the direction in which the force is applied. The force produced by the angular momentum of the rotating body and the applied force results in this precessional motion. Gyroscopes are used in aircraft, ships, submarines, rockets and in many other auto-navigation type vehicles.
The transferal of angular momentum into energy and energy into angular momentum has led to numerous advances in technologies over the last century. Converting the linear kinetic energy stored in wind to angular momentum which was used to run windmills (and running water in the case of water mills) greatly aided in the progression of the industrial revolution. This principle is still in use today, at Niagra Falls, where water runs over turbines making them spin which turn their rotational energy into electricity which we use to power our homes. Rotational kinetic energy can be described as follows: Kerot= 1/2 mv2 , but recall that v = r? so Kerot = 1/2 mr2?2 however I = mr2 ? Kerot = 1/2 I?2
These formulas allow one to follow the transferal of rotational energy to and from linear or other forms of energy.
Angular momentum is used to explain many things, and it is has many applications. Angular momentum is also essential to our very existence, without the conservation of angular momentum we might drift into the sun or away into space. Angular momentum is a very important part of physics and physics is a very important part of angular momentum.
ENDNOTES
- Raymond A. Serway, Physics For Scientists and Engineers, (Toronto: Saunders College Publishing, 1996) p. 325.
- David G. Martindale, Fundamentals of Physics: A Senrior Course, (Canada: D.C. Heath Canada Ltd., 1986) p. 320.
- ibid
- Raymond A. Serway, Physics For Scientists and Engineers, (Toronto: Saunders College Publishing, 1996) p. 325.
Bibliography
Blott, J. Frank, Principles of Physics: Second Edition Publisher not given: 1986
David G. Martindale., Fundamentals of Physics, Canada: D.C. Heath Canada Ltd. 1986
Olenick, P. Richard, The Mechanical Universe: Introduction to Mechanics and Heat, Cambridge: Cambridge University Press 1985
