You want to determine the thickness x of the ring, such that when two such concentric rings are taped together, the mass of the disk will be identical to the mass of the pair of concentric rings. Figure 1 shows a diagram with two such objects. The other object will be a ring whose outer radius is also 3.5", but whose mass is identical to the mass of the disk. One of the objects will be a circular disk of radius 3.5". You will construct two planar objects from card stock paper. You will begin by making use of your maker skills as well as some basic math skills, including algebra and geometry. In this lesson you will be comparing the moments of inertia of two objects with identical masses. Task #1 - Prepare Planar Disk and Ring with Identical Masses It should be noted that by the definition, the moment of inertia of a body depends not only on the particular axis on which it rotates, but also on its shape and the way in which its mass is distributed. For continuous rigid objects, the equation would be similar but making use of integrals instead of a sum.
The moment of inertia is the sum of the masses of the particles making up the object multiplied by their respective distances squared from the axis of rotation. Moment of inertia can be defined by the equation This is expressed by the equation τ = Iα . By analogy, the moment of inertia of any rigid object is a measure of its resistance to angular acceleration about an axis when a torque is applied to the object.
Newton's Second Law of Motion expresses this in the familiar equation F = ma. In addition to being a property of any physical object, mass is a measure of the resistance of an object to acceleration when a net force has been applied to the object. At the heart of these comparisons lie the concepts of mass on one hand and moment of inertia on the other. There are numerous analogies when comparing linear and rotational motion.
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