Hanging Sign Problem: Find Angle & Distance from Ceiling

[KatieLynn]

Homework Statement

If you are hanging a sign from the ceiling with two wires, each with a safe working load of 20N. The hooks at the top of the sign are 2.0 meters apart while the hooks in the ceiling are 4.0 meters apart. A) If the sign weighs 30N what is the minimum angle to the horizontal the wires can be to safely hang the sign, and B) how far from the ceiling is the top of the sign?

Homework Equations

F=MA
I'm not sure if that it or not... but that's my best guess.

The Attempt at a Solution

I attached the picture(s) I drew to attempt to solve it. I'm not sure what angle to solve for or how to solve for it.

 

[Astronuc]

OK, the tension in one line must be less than 20 N, and each line must bear half the weight (through symmetry). Half the weight is 15 N, if the line was vertical, but since the upper fixtures are 4 m apart and the fixtures on the sign are 2 m apart, then the top of line is offset by 1 m from the bottom of the line, so the line is at an angle.

With the line at an angle, the tension in the line must increase since the vertical component must be 15 N, and the tension must never exceed 20 N. What angle achieves that?

With that angle, what is the length of line for a horizontal displacement of a 1 m (horizontal leg/base) of a right triangle?

 

[ogb p]

I would approach this problem by first identifying the relevant equations and principles that can be applied. In this case, the two wires supporting the sign can be modeled as a system of forces acting on the sign. The sign's weight of 30N can be represented as a downward force acting on the center of mass of the sign.

To find the minimum angle to the horizontal that the wires can be, we can use the concept of equilibrium, where the sum of all forces acting on the sign must be equal to zero. In this case, the two wires must be able to support the weight of the sign and balance each other out. This can be represented mathematically as:

ΣF = 0

where ΣF represents the sum of all forces acting on the sign. We can also break down this equation into its x- and y-components:

ΣFx = 0
ΣFy = 0

In the x-direction, we have two forces acting on the sign - the horizontal components of the two wires. Since we are looking for the minimum angle, we can assume that the wires are at an equal angle to the horizontal. This means that the horizontal components of the two wires must be equal and opposite to each other. Therefore, we can write the equation as:

Fwx = -Fwy

where Fwx and Fwy represent the horizontal components of the wires. We can also use the trigonometric relationship between the angle and the horizontal component of the force, which is given by:

Fwx = Fwcosθ

where θ is the angle between the wire and the horizontal. Substituting this into our equation, we get:

Fwcosθ = -Fwcosθ

Solving for θ, we get:

θ = cos^-1 (1/2)

θ = 60°

Therefore, the minimum angle to the horizontal that the wires can be is 60°.

To find the distance from the ceiling to the top of the sign, we can use the concept of torque, which is the tendency of a force to rotate an object about a fixed point. In this case, we can use the principle of moments, which states that the sum of the moments of all forces acting on an object must be equal to zero. Mathematically, this can be written as:

ΣM = 0

Again, we can break down this equation into its x- and