Find the center of mass of a uniform plate shown in Fig. 7.10a.

Answer

The centre of mass is at a distance of 4.3 units from the left hand edge of the plate.

Explanation

Since the plate is uniform, the mass of any part of it is proportional to its area. Moreover, noting that the x-axis, directed as shown in the figures, is the axis of symmetry of the plate, we may conclude that the cen- tre of mass lies on this axis, i.e. that `Y_(e) = z_(c) = 0`. Then the solution may be obtained by two methods. First method. Imagi_ne the plate cut in two- into a triangle and a symmetrical remaining piece (Fig. 7.10b). The centre of mass of the· triangle lies at a distance of one third of a median from the origin, i.e . `x_(1) = 1`, the centre of mass of the second body lies at its centre of symmetry, i.e. at a distance `x_(2) = 5`

from the origin. The masses of those bodies are `m_(1) = 6 xx 3//2 = 9` and `m_(2) = 6 xx 10 - 2 xx 9 = 42` conventional units. The coordi- nate of the centre of mass is
`x_(c)=(m_(1)x_(1)+m_(2)x_(2))/(m_(1)+m_(2))=(9xx1+42xx5)/51=4.3`
Second method. Consider the plate as a sum of two bodies, a rec- tangle of mass `m_(3) = 60` units and a triangle of negative mass `m_(1) = -9` units (Fig. 7.10c). The coordinates of their centres of mass are `x_(3) = 5` and `x_(4) = 9`. We have according to the definition
`=(m_(3)x_(3)+m_(4)x_(4))/(m_(3)+m_(4))=(60xx5-9xx9)/(60-9)=4.3`
Naturally, both methods produce the same result.