 #1
SilasHokanson
 4
 0
 Homework Statement:

Given a perfectly inelastic tetrahedron in 3space with coordinates:
a = < a1, a2, a3 >
b = < b1, b2, b3 >
c = < c1, c2, c3 >
d = < 0, 0, 0 >
Given a force vector F is applied to point D
Assume points A, B and C are fixed
Assume tetrahedron is completely still (is in static & rotational equilibrium)
Calculate the normal forces on points A, B and C in order to maintain said equilibriums
 Relevant Equations:

X is Cross Product
Torque_Net = 0
Force_Net = 0
Torque = Force X Distance
My approach to this problem is to recognize that the tetrahedron being still means that net torque is zero and net force is zero.
Fd is given
Fa + Fb + Fc = Fd
Fa X a + Fb X b + Fc X c = <0,0,0>
This can be split up into a series of 6 equations, 2 for each component.
However, this is where I get stuck. 2 equations for each component is not enough to solve each system, since solving a system of 3 variables (x,y,z) necessitates 3 equations?
This is not technically a homework problem, it is a problem I posed myself, however I am convinced it must be solvable, if I imagine the situation in my head. A tetrahedron with 3 of its faces fixed, and a load applied to the fourth cannot move, since fixing 3 points is enough to determine an objects position in 3 space. Therefore there must be a definite normal force being applied on each of the 3 vertices, in terms of all variables given in this problem?
I have also attempted solving this problem generally, as in using actual numbers for some positions instead of variables, but it doesn't make the problem any less difficult.
I may be in over my head here, or I may just be missing some critical information. I'm fairly certain a third "value" that I could set to zero would be enough to solve this problem? Or perhaps I'm just setting it up incorrectly.
Fd is given
Fa + Fb + Fc = Fd
Fa X a + Fb X b + Fc X c = <0,0,0>
This can be split up into a series of 6 equations, 2 for each component.
However, this is where I get stuck. 2 equations for each component is not enough to solve each system, since solving a system of 3 variables (x,y,z) necessitates 3 equations?
This is not technically a homework problem, it is a problem I posed myself, however I am convinced it must be solvable, if I imagine the situation in my head. A tetrahedron with 3 of its faces fixed, and a load applied to the fourth cannot move, since fixing 3 points is enough to determine an objects position in 3 space. Therefore there must be a definite normal force being applied on each of the 3 vertices, in terms of all variables given in this problem?
I have also attempted solving this problem generally, as in using actual numbers for some positions instead of variables, but it doesn't make the problem any less difficult.
I may be in over my head here, or I may just be missing some critical information. I'm fairly certain a third "value" that I could set to zero would be enough to solve this problem? Or perhaps I'm just setting it up incorrectly.