Daniel B. answered • 07/03/22
A retired computer professional to teach math, physics
The brick has two faces that are a×a squares, and four faces that are a×2a rectangles.
I am assuming that it is resting on one of the rectangular faces, and it is to be positioned
on one of its square faces.
I am going to assert without proof that the minimal amount of work will be needed
by turning it along one of the shorter edges it is resting on.
The amount of work needed is the difference between the brick's initial energy and
its final energy or its maximal energy; see the discussion below.
The brick's mechanical energy is the sum of its potential and kinetic energy.
We minimize that sum by moving the brick so slowly that its kinetic energy
will be negligible in comparison to its potential energy difference.
From this point on we talk only about its potential energy.
In calculating the potential energy we can treat the brick as if its entire mass
were concentrated at its center of gravity.
The mass is the product of its density, p, and its volume:
m = 2a³p
(Let g = 9.81 m/s² be gravitational acceleration).
At the outset the center of gravity is at height a/2 from the supporting surface,
at the end it is at height a from the supporting surface, and
during the rotation its maximal distance from the supporting surface is a√5/2.
Consequently the initial potential energy is mga/2,
its final potential energy is mga,
and its maximal potential energy is mga√5/2.
During the rotation we must exert work mga√5/2 - mga/2 = mga(√5-1)/2
to bring the brick into its highest position standing on its edge.
After that the brick falls into its vertical position, for which we exert negative work.
If you are to count this negative work, then the total work is
mga - mga/2 = mga/2
However, normally you are unable to utilize the energy of the brick falling from its
maximal height to its vertical position, so it is wasted into heat.
In that case your work is mga(√5-1)/2, which brings the brick into its vertical
position plus generates some heat.
(I am sorry, I cannot tell which of the two interpretations you are being asked to assume.)
