A massive m is placed on a stick of length r and negligible mass, and constrained to rotate about a solved axis. If the mass is released from a horizontal orientation, it can be described either in regards to force and also accleration with Newton"s 2nd law for linear motion, or as a pure rotation about the axis through Newton"s 2nd law for rotation. This gives a setup for comparing linear and also rotational quantities for the very same system. This process leads come the expression because that the minute of inertia the a suggest mass.

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For a uniform rod v negligible thickness, the moment of inertia around its center of fixed is

For mass M = kg and length together = m, the minute of inertia is ns = kg m²Show |

The minute of inertia around the finish of the rod isI = kg m².Show |

If the thickness is no negligible, climate the expression for i of a cylinder about its end can be used.

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Calculating the moment of inertia that a rod about its facility of fixed is a good example that the need for calculus to resolve the nature of consistent mass distributions. The minute of inertia the a point mass is provided by ns = mr2, however the rod would have to be taken into consideration to be an infinite number of point masses, and also each need to be multiplied by the square the its street from the axis. The resulting infinite sum is dubbed an integral. The general form for the minute of inertia is:

When the mass facet dm is express in terms of a length facet dr along the rod and also the amount taken over the whole length, the integral take away the form:

display details |

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The minute of inertia calculation for a uniform rod requires expressing any mass element in terms of a distanceelement dr along the rod. To carry out the integral, the is essential to express eveything in the integral in terms of one variable, in this instance the length variable r. Because the totallength L has mass M, climate M/L is the proportion of mass to length and the masselement deserve to be expressed together shown. |

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when the moment of inertia of one object around its facility of mass has beendetermined, the moment around any other axis deserve to be determined by the Parallel axis theorem: |

In this case that becomes

This deserve to be evidenced by directintegration

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