Expected Moment

Classical Mechanics Level 4

A uniform rod of mass M M and length L L is spun about a perpendicular axis passing through a random point along its length. If the expected moment of inertia about the random axis is α M L 2 \alpha M L^2 , determine the value of α \alpha .

Details and Assumptions: The probability distribution is uniform as a function of length

Bonus: How does the answer relate to other well-known rod moments?


The answer is 0.1666667.

Steven Chase by Steven Chase , 37, USA

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1 solution

Steven Chase
Nov 10, 2017

Parallel axis theorem (and its application to a rod of length L L whose center is at x = 0 x=0 ):

I = I C M + M d 2 I = M L 2 12 + M x 2 I = I_{CM} + M d^2 \\ I = \frac{M L^2}{12} + M x^2

Take a length-weighted average of the moments by integrating from x = − L 2 x = -\frac{L}{2} to x = L 2 x = \frac{L}{2} and dividing by the total length:

I a v = 1 L ∫ − L / 2 L / 2 ( M L 2 12 + M x 2 ) d x = 2 ( M L 2 12 1 2 + M 3 L 2 8 ) = M L 2 6 I_{av} = \frac{1}{L} \int_{-L/2}^{L/2} \Big(\frac{M L^2}{12} + M x^2 \Big) dx = 2 \Big(\frac{M L^2}{12} \frac{1}{2} + \frac{M}{3} \frac{L^2}{8}\Big) = \frac{ML^2}{6}

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Nice question!

Md Zuhair - 3 years, 6 months ago

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Thanks...............................

Steven Chase - 3 years, 6 months ago

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