CLASS2 - 1 CHAPTER 2 MOMENT OF INERTIA 2.1 Definition of...

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1 CHAPTER 2 MOMENT OF INERTIA 2.1 Definition of Moment of Inertia Consider a straight line (the "axis") and a set of point masses K , , , 3 2 1 m m m such that the distance of the mass m i from the axis is r i . The quantity 2 i i r m is the second moment of the i th mass with respect to (or "about") the axis, and the sum 2 i i r m is the second moment of mass of all the masses with respect to the axis. Apart from some subtleties encountered in general relativity, the word "inertia" is synonymous with mass - the inertia of a body is merely the ratio of an applied force to the resulting acceleration. Thus 2 i i r m can also be called the second moment of inertia . The second moment of inertia is discussed so much in mechanics that it is usually referred to as just "the" moment of inertia. In this chapter we shall consider how to calculate the (second) moment of inertia for different sizes and shapes of body, as well as certain associated theorems. But the question should be asked: "What is the purpose of calculating the squares of the distances of lots of particles from an axis, multiplying these squares by the mass of each, and adding them all together? Is this merely a pointless make-work exercise in arithmetic? Might one just as well, for all the good it does, calculate the sum i i r m 2 ? Does 2 i i r m have any physical significance?" 2.2 Meaning of Rotational Inertia . If a force acts of a body, the body will accelerate. The ratio of the applied force to the resulting acceleration is the inertia (or mass) of the body. If a torque acts on a body that can rotate freely about some axis, the body will undergo an angular acceleration. The ratio of the applied torque to the resulting angular acceleration is the rotational inertia of the body. It depends not only on the mass of the body, but also on how that mass is distributed with respect to the axis. Consider the system shown in figure II.1. FIGURE II.1
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2 A particle of mass m is attached by a light (i.e. zero or negligible mass) arm of length r to a point at O, about which it can freely rotate. A force F is applied, and the mass consequently undergoes a linear acceleration F/m . The angular acceleration is then F/ ( mr ) . Also, the torque is Fr . The ratio of the applied torque to the angular acceleration is therefore mr 2 . Thus the rotational inertia is the second moment of inertia. Rotational inertia and (second) moment of inertia are one and the same thing, except that rotational inertia is a physical concept and moment of inertia is its mathematical representation. 2.3 Moments of inertia of some simple shapes . A student may well ask: "For how many different shapes of body must I commit to memory the formulas for their moments of inertia?" I would be tempted to say: "None". However, if any are to be committed to memory, I would suggest that the list to be memorized should be limited to those few bodies that are likely to be encountered very often (particularly if they can be used to determine quickly the moments of inertia of other bodies) and for which it is easier to remember the formulas than to derive them.
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CLASS2 - 1 CHAPTER 2 MOMENT OF INERTIA 2.1 Definition of...

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