A wooden hammer is shown above. The dimensions are as follows: The dimensions are as follows: a = 10 cm, b = 8 cm, c = 18 cm, d = 40 cm, e = 3 cm. Determine the centroid of the hammer.

We start by noting that the hammer is symmetric about the y-z plane and the x-y plane, which means that the centroid is along the y axis. The means that the centroid in the x-direction and the centroid in the z-direction is zero.

This leaves the y-direction. We proceed as follows:

  • The volume of the head is V1 = (a)(b)(c) = (10)(8)(18) = 1440 cm2.
  • The centroid of the head is halfway into the y thickness of the head, thus y1 = 4 cm.
  • The centroid of the handle is at the centre of the parallelepiped, thus it is 20 cm from where it connects with the head and another 8 cm to the origin, thus the centroid of the handle y2 = 28 cm.
  • The volume of the handle is V2 = (d)(e)(e) = (40)(3)(3) = 360 cm2.

The y-centroid is thus computed by the formula

\bar{y} = \frac{V_1\times y_1 + V_2 \times y_2}{V_1 + V_2}

Substituting yields \bar{y} = 8.8\,cm .

The problem is taken from Movnin and Izraelit (1970), where the authors dispense with the laziness exhibited above and actually compute the centroids in the x- and z-directions.