If a solid semicylindrical block (a block-letter D) is placed on its curved side on a horizontal table, the top (the flat side) will be parallel to the table.

The table is tilted by an angle, a, perpendicular to the axis of the cylinder. What angle will the top of the block form with the table now?

Assume angle a is small enough to prevent the block from slipping or tipping over.

Notation looking at cross section of "D" shape
perpendicular to its axis:

 A: Center of the "D" shape
 B: Point of contact between the "D" shape and
    the table top
 C: Center of mass of the "D" shape
 a: Angle between the table top and horizontal
 b: Desired angle between top of the "D" shape
    and the table top
 h: |CA|/|BA| = 4/(3*PI)

Let D be the intersection of the ray AC and the
line through B parallel to the top of the "D"
shape. Then

 |BA| sin(b) = |BA| sin(BAD) = |BD| = |BC| sin(BCD) = |BC| sin(a+b)

 |BC| sin(a) = |CA| sin(b)

Combining these two we get

 sin(a+b) = sin(a)/(|CA|/|BA|) = sin(a)/h

                or

 b = arcsin(sin(a)/h) - a

Edited on January 17, 2006, 10:59 am

Posted by Bractals on 2006-01-17 01:43:39