A uniform wooden plank of length and mass is to be installed as a drawbridge

PHYSICS · Multi-part · A Level · Physics · Engineering physics · Engineering physics

Problem

A uniform wooden plank of length

L

and mass

M

is to be installed as a drawbridge. Engineers may place the pivot axis anywhere along the plank, but regulations stipulate that the rotational inertia about the chosen axis must not exceed

1.5\,I_{\text{cm}}

, where

I_{\text{cm}}

is the plank’s rotational inertia about an axis through its center and perpendicular to the plank. (For a uniform rod about such a central axis,

I_{\text{cm}}=\tfrac{1}{12}ML^{2}

.) (a) Without inserting numerical values, use algebra to obtain an inequality that limits the allowable distance

d_{\max}

between the pivot axis and the plank’s center of mass. (b) Express

d_{\max}

explicitly in terms of

L

. (c) Describe qualitatively how adding extra mass near the ends of the plank (while keeping the pivot location fixed) would affect the maximum allowable pivot distance found in part (b). Support your explanation with physical reasoning about mass distribution.

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