How Do You Calculate Tensions T1 and T2 in a Balanced Suspended Plank?

[tzar1990]

Homework Statement

The object is balance. Calculate T₁ and T₂. (yes, this plus the diagram is really the entirety of the question)

Homework Equations

Στ = 0
τ = F⋅r
F = mg

The Attempt at a Solution

First, we treat the left side as a fixed point and solve for the vertical component of T₂

Στ = 0
20⋅9.8⋅1 + 10⋅9.8⋅1.5 - 2T_2y = 0
343 = 2T_2y
T_2y = 171.5

Next, we treat the right side as fixed and solve for the vertical component of of T₁
Στ = 0
0.5⋅10⋅9.8 + 1⋅20⋅9.8 - 2⋅T_1Y = 0
245 = 2T_1Y
T_1Y = 122.5

To verify, we check that the sum of the forces up should equal the sum of the forces down
T_1Y + T_2Y = 20*9.8 + 10 * 9.8
171.5 + 122.5 = 196 + 98
294 = 294

And this is the point where I get stuck. As far as I can tell, the situation is balanced for all cases where the force left due to T₁ is equal to the force right due to T₂, meaning you have everything from the case where both wires are vertical (and thus the x-components of their tension is zero) to the case where both are nearly horizontal (and the x-components of both approach infinity) being true.

What am I missing?

 

[SteamKing]

Unless you know the angles which T1 and T2 make with the horizontal, you can't work out the horizontal components of the tension.
Writing the moment equation can only give you the vertical component of the tensions.

 

[tzar1990]

Thank you! That's what I told my student, but she seemed doubtful that an official question would have a misprint like that. Nice to hear that I'm not just missing something obvious.

 

[haruspex]

Unless you know the angles which T1 and T2 make with the horizontal, you can't work out the horizontal components of the tension.

... except that, knowing one angle would do.

 

[HalfLostAndSearching]

I would suggest that you consider the physical constraints of the problem. While the equations you have used are correct, they only represent a mathematical solution. In reality, the plank cannot be in a state where the tension in both wires is zero or infinity. Therefore, the solution you have obtained may not be physically feasible.

You may need to consider other factors such as the weight and dimensions of the plank, the strength and elasticity of the wires, and any other external forces acting on the system. These factors could affect the equilibrium state of the plank and the tension in the wires. It may also be helpful to draw a free body diagram for each wire and consider the forces acting on them separately.

Additionally, it is important to note that the torque equation you have used assumes that the forces are acting at a perpendicular distance from the pivot point. In this case, the forces are acting at an angle, so you may need to use trigonometry to calculate the perpendicular components of the forces.

Overall, as a scientist, it is important to always consider the physical constraints and real-world implications of a mathematical solution. It may be necessary to make assumptions or simplifications in order to obtain a practical and feasible solution.