Suppose we have a gate in a fence. It is attached by two hinges at the corners of one side, and the opposite upper corner is also connected to the post by a cable at 30° from the horizontal. The gate has a mass 20 kg, distributed so that its center of mass is at its geometric center; and is 120 cm tall and 90 cm wide.
Suppose that the hinges can only withstand a horizontal force component of less than 60 N in either direction. Within what range must the tension in the cable lie?
I will perform the solution with general variables, and then input the specific values of the problem. Let us refer to the force on the gate by the upper hinge as , that due to the lower hinge as
, and that by the tension in the cable as
, which is at an angle θ
The gate has mass M, width W, and height H.
We have a static equilibrium, so the net forces and torque are zero. We choose to have the signs on the components of the forces so that they are positive if as in figure 2. Looking at horizontal forces, we have
, (1)
and for vertical forces, we have
. (2)
Now, let us consider the torque about the center of mass:
. (3)
We can solve these three for and
:
We can rewrite eq. 3 as
. (4)
Multiplying eq. 1 by H and adding to both sides:
Now, we can solve eq. 2 for and substitute into the above to get:
.
Similarly, we can Multiply eq. 1 by H and subtract it from both sides of eq. 4:
Now, using our values M=20 kg, g=9.8 m/s2,
W=90 cm, H=120 cm, and θ=30°, we have
and
.
Our requirements are that and
. The first of these gives:
;
and the second gives us
.
Combining, we see that the acceptable cable tension range is .
Tags: Forces, Friday Physics, physics, Statics
