- #1
Rasoul
- 5
- 0
I have studied a few online sources about static equilibrium in a mechanical system. My overall understanding is that for an object to be in static equilibrium, two following conditions have to be fulfilled:
1) Vector sum of all external forces that act on body must be zero.
2) Vector sum of all external torques that act on the body, measured about any possible point must be zero.
I fully understand condition (1), but the statement of condition (2) is not clear for me. Let's say that an object A is supported by two other objects, B and C, where each contact point, (A,B) and (A,C) could be a possible rotation axis for A. So, I can write 9 equations for unknown reaction forces:
set 1) F_{total, x} = 0, F_{total, y} = 0, F_{total, z} = 0
set 2) M_{total, (A,B), x} = 0, M_{total, (A,B), y} = 0, M_{total, (A,B), z} = 0
set 3) M_{total, (A,C), x} = 0, M_{total, (A,C), y} = 0, M_{total, (A,C), z} = 0
But we know that there exist only 6 independent equations. My question is that, if I use (set 1) and (set 2) to solve for unknown forces (I assume that the system has a solution), then can I deduce that the object A will also have zero torque about (A,C)? In other words, does a solution for (set 1) and (set 2) imply that that solution is also a solution for (set 1) and (set 3)?
I have found a case in which, all three objects are cuboids and the contact point (A,B) is a line segment (an edge of B on a face of A) but the contact point (A,C) is a single point (a vertex of C on a face of A). In this case, the solution of (set 1) and (set 2) is not a solution for (set 1) and (set 3).
1) Vector sum of all external forces that act on body must be zero.
2) Vector sum of all external torques that act on the body, measured about any possible point must be zero.
I fully understand condition (1), but the statement of condition (2) is not clear for me. Let's say that an object A is supported by two other objects, B and C, where each contact point, (A,B) and (A,C) could be a possible rotation axis for A. So, I can write 9 equations for unknown reaction forces:
set 1) F_{total, x} = 0, F_{total, y} = 0, F_{total, z} = 0
set 2) M_{total, (A,B), x} = 0, M_{total, (A,B), y} = 0, M_{total, (A,B), z} = 0
set 3) M_{total, (A,C), x} = 0, M_{total, (A,C), y} = 0, M_{total, (A,C), z} = 0
But we know that there exist only 6 independent equations. My question is that, if I use (set 1) and (set 2) to solve for unknown forces (I assume that the system has a solution), then can I deduce that the object A will also have zero torque about (A,C)? In other words, does a solution for (set 1) and (set 2) imply that that solution is also a solution for (set 1) and (set 3)?
I have found a case in which, all three objects are cuboids and the contact point (A,B) is a line segment (an edge of B on a face of A) but the contact point (A,C) is a single point (a vertex of C on a face of A). In this case, the solution of (set 1) and (set 2) is not a solution for (set 1) and (set 3).