Moment of inertia
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Moment of Inertia
Recall that the position, P, of an object on a circular track of radius r can be specified by the angle, ?, made with a chosen axis: [Diagram goes here - download the original to see it.] If the object is moving around the track it will have an angular velocity - the rate of change of ? with t. [Equation goes here - download the original to see it.] If the object is accelerating around the track it will have an angular acceleration: [Equation goes here - download the original to see it.] Forces cause objects to accelerate, as indicated by Newton's 2nd law. When a force acts on an object at a distance from its centre of mass it also causes that object to spin. The effect of this is called torque of moment. Just as forces causes objects to accelerate so torques causes angular acceleration - torque causes a change in angular velocity. [Diagram goes here - download the original to see it.] Newton's Second Law for angular acceleration states: [Equation goes here - download the original to see it.] We can compare this with Newton's Second Law; [Equation goes here - download the original to see it.] Then Newton's Second Law leads to the equation, F = ma. Here mass, m, is the constant of proportionality in Newton's Second law. Consequently, mass is more appropriately termed 'inertial mass'. Designating torque by C and angular acceleration by [Equation goes here - download the original to see it.] we seek an equivalent constant of proportionality for the angular application of Newton's Second Law. The constant, designated I, is called the moment of inertia.. Torque = moment of inertia [Equation goes here - download the original to see it.] angular acceleration [Equation goes here - download the original to see it.] Just as inertial mass is a property of every physical body, so moment of inertia is a property if every physical body. The moment inertia must be derived with respect to an axis of rotation. A standard axis will be one that passes through the centre of mass of an object, and, if that object is a rod or a plane lamina, will pass perpendicular to that rod or lamina. [Diagram goes here - download the original to see it.] 1. A particle is a null dimensional object of mass M but no size. It can have a moment of inertia about any axis at a distance r. 2. A rod is a one-dimensional object of theoretically no radius. Any standard axis of rotation passes through the centre of mass c, and is perpendicular to the mass. 3. A lamina is a two-dimensional object of theoretically no breadth. The standard axis of rotation passes through the centre of mass and is perpendicular to the surface. 4. A solid is a three-dimensional object. The standard axis of rotation passes through the centre of mass. However, only solids with an axis of symmetry have an axis of rotation. For asymmetrical solids the moment of inertia is different for each axis of rotation. We seek a definition of the moment of inertia it terms of its mass and location of its axis of rotation. We seek rules for the addition of moments of inertia. When two particles are joined together we merely ass their masses to find the inertial mass, but we cannot expect that addition of moments of inertia will obey such simple laws!
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Contents of
Moment of inertia
1 Moment of Inertia
2 Definition of inertia for a particle
3 Addition law for moments of inertia for two or more particles rotating about the same axis of rotati
4 Standard results for moments of inertia
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