This is an example of three-dimensional equilibrium using couples. The problem comes from Theoretical Mechanics – A Short Course -Targ.

Consider the cube at the right, and the following conditions:

  • Force pair P and P’ produce couple m1
  • Force pair Q and Q’ produce couple m2
  • Force pair N and N’ produce couple m
  • Coordinates x and y shown, z is upward from O
  • Neglect weight of cube (sides are equal, length a)

Find values of P and Q so that system is in equilibrium, and Value of N (reactions.)

We proceed as follows:

  • Vectors for moments:
    • m1 = m1 j
    • m2 = m2 i
    • m = -.7071 i – .7071 j
  • Summing moments:
    • X-direction = m2 – .7071 m = 0
    • Y-direction = m1 – .7071 m = 0
  • We can see from the above that both m1 and m2 are both equal to .7071 m, and are thus equal to each other
  • Since m1 = P a and m2 = Q a, P and Q are equal to each other for equilibrium
  • Substituting x-direction results, Q a = .7071 m
  • From the geometry, m = 1.414 a N
  • Substituting, Q a = P a = N a and thus Q = P = N