A carpenter cuts a thin wooden board into the shape of an isosceles trapezoid

MATHEMATICS · Multi-part · ISC · Mathematics · Physics · Calculus

Problem

A carpenter cuts a thin wooden board into the shape of an isosceles trapezoid. The trapezoid has a lower base of length

2a

, an upper base of length

b

(

b<2a

), and a vertical height

h

. The wood is uniform in thickness and density. A coordinate system is chosen with the origin at the midpoint of the lower base, the

x

–axis along that base, and the

y

–axis vertical (positive upward). (a) Using symmetry, state and justify the

x

–coordinate of the trapezoid’s center of mass. (b) Derive an expression for the

y

–coordinate of the center of mass in terms of

a

b

, and

h

by decomposing the trapezoid into simpler geometric shapes of your choice. (c) Evaluate your expression numerically for

a = 0.40\,\text{m}

b = 0.20\,\text{m}

, and

h = 0.30\,\text{m}

. Present the final answer with appropriate units.