Use equation I=∫r2dm to calculate the moment of inertia of a slender, uniform rod with mass M and length L about an axis at one end, perpend

Question

Use equation I=∫r2dm to calculate the moment of inertia of a slender, uniform rod with mass M and length L about an axis at one end, perpendicular to the rod. Express your answer in terms of the variables M and L.

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Thu Thảo 3 years 2021-07-29T16:28:41+00:00 1 Answers 358 views 0

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    1
    2021-07-29T16:29:54+00:00

    Answer:

    The moment of inertia is    I= \frac{1}{3} ML^2

    Explanation:

    From the question we a given an equation

                            I = \int\limits{r^2} \, dm

    Now let look at a small length da of this uniform rod that a meters form the x-axis

     The mass of this small section can be mathematically evaluated as

                           dM = da *\frac{M}{L}

    For this small portion for the rod the moment of inertia is

                       dI = (dM)a^2

    Substituting for dM

                      dI = da [\frac{M}{L} ] a^2

    to get I we integrate both sides

                     I = \int\limits^L_0 {\frac{M}{L} * a^2} \, da

                       = [\frac{M}{L} ][\frac{a^3}{3} ]\left L} \atop {0}} \right.

                       I= \frac{1}{3} ML^2

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