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Well Balanced Letter: V (Posted on 2007-05-07) Difficulty: 3 of 5
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A nice letter V is crudely shown above.
The goal is to get its center of gravity to be at the exact point where the two inner diagonal segments meet.

Case 1: Find the width of the V if the two horizontal segments are one unit long and the overall height of the V is three units.

Case 2: The width and the height are each one unit. Find the lengths of the horizontal segments.

No Solution Yet Submitted by Jer    
Rating: 4.0000 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution Comment 1 of 1
Since the horizontal centroid is obvious the only question is the vertical.

Cut the figure into top and bottom.

define:
w width of "cut out" triangle
x width of bottom triangle (2 * width of horiz segment)
y height of the "cut out" triangle
z height of bottom triangle

Note that given any angle, the cut out triangle and the bottom triangle are similar (w/x) = (y/z).  Also, the top "wings" are equivalent to a rectangle of width x and height y wrt vertical centroid.

The centroid of the bottom is at z/3 down (triangle) and the top is y/2 up (rectangle).
The area of the bottom is xz/2 and the top is xy

so, for the center to be at the cut, (z/3)(xz/2) = (y/2)(xy)
(1/3)z^2=y^2
z = y√3
by similarity, x = w√3

Case 1 : width = 2+w = 2+2/√3 = 2(1+1/√3)
(note that the height was not needed)

Case 2 : 1 = x+w = x+x/√3 = x(1+1/√3) = x(3+√3)/3
x = 3/(3+√3)
horiz segment = x/2 = (3/2)/(3+√3)

Once again, height not needed.

-- Joel

  Posted by Joel on 2007-05-07 15:19:44
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