Equilibrium of bodies III
Equilibrium of bodies III: We now move on to three dimensional (3-d) cases. In three dimensional cases the equilibrium conditions lead to balance along all three axes. Then
We now have to take care of components of forces and torque in all three dimensions. The engineering elements that we considered earlier are now considered as 3-d case. Thus consider a ball-socket joint in which a ball is supported in a socket (figure 1).
A ball-socket joint provides reaction forces Nx, Ny and Nz in all three directions (figure 1) but it cannot apply any torque. This is a little like a hinge joint in 2d . Let me solve an example using such a joint.
Example 1: To balance a heavy weight of 5000 N, two persons dig a hole in the ground and put a pole of length l in it so that the hole acts as a socket. The pole makes an angle of 30° from the ground. The weight is tied at the mid point of the pole and the pole is pulled by two horizontal ropes tied at its ends as shown in figure 2. Find the tension in the two ropes and the reaction forces of the ground on the pole.
To solve this problem, let me first choose a co-ordinate system. I choose it so that the pole is over the y-axis in the (y-z) plane (see figure 2).
The ropes are in (xy) direction with tension T in each one of them so that tension in each is written as
You may be wondering why I have taken the tension to be the same in the two ropes. Actually it arises from the torque balance equation; if the tensions were not equal, their component in the x-direction will give a nonzero torque.
Let the normal reaction of the ground be (Nx , Ny , Nz). Then the force balance equation gives
Taking torque about point O and equating it to zero, we get