Support Wire Connected to Sign

Question

Hang a metal sign from a single vertical wire. The sign has a triangular shape with a base of width ‘a’, a height of ‘b’, and a hypotenuse of length ‘c’, and is oriented in an x-y plane such that the corners are at the following points: (0,0), (a,0), (a,b). The bottom of the sign is to be parallel to the ground. At what distance, x, from the left end of the sign should you attach the support wire?

Answer 1

Find:

At what distance, x, from the left end of the sign should you attach the support wire?

Solution:

1. Weight always acts through the center of gravity of an object, which represents the average location of the weight of an object.
2. The cg of the triangular metal sign is located 2𝑎/3 to the right of the origin and at a distance of 𝑏/3 above the origin.
3. In order for the metal sign to achieve static equilibrium, the sum of the forces and moments should equal zero.
4. Draw the free body diagram of the metal sign.
5. Assuming the downward direction is negative and the upward direction is positive, apply the sum of the forces in the y-coordinate direction.
6. ∑Fy=0=T-W
7. T=W
8. Consequently, the magnitude of the tension (T) and weight (W) forces are equivalent.
9. Assuming clockwise is the positive direction, apply the sum of the moments about the origin in order to solve for the distance, x.
10. ∑Mo=0=Wx-T(2a/3)
11. Wx=T(2a/3)
12. Substitute T for W in order to cancel the T terms and solve for x.
13. Tx=T(2a/3)
14. x=2a/3
15. Hence, the support wire should be attached at a distance x=2𝑎/3 to the right of the origin in order for the metal sign to be in static equilibrium.