How Do You Solve the Moving Pulley Problem with a 100 kg Sack for Equilibrium?

[Jay9313]

Homework Statement

http://media.cheggcdn.com/media/76d/76dd5c56-43a5-44d3-b8b8-40ba63a9010a/phpYYNJyN.png

Cable ABC has a length of 5 m. Determine the position x and the tension developed in ABC required for equilibrium of the 100 kg sack. Neglect the size of the pulley

Homework Equations

T_ABCosθ + T_BCCos$\phi$=0
T_ABSinθ + T_BCSin$\phi$=980N
I also have the distance from C to the pulley (in x direction) is 3.5-x

The Attempt at a Solution

My attempt at the solution so far is
T_ABCosθ = T_BCCos$\phi$
I can rearrange the trig functions themselves, but I'm missing a height component, but I don't even know if that's the right track.

 

[Jay9313]

I have also managed to introduce a few new variables. I have introduced h+H=5 (The length of the rope) and y and y+0.75

 

[haruspex]

Consider the 'free body' diagram for a thin vertical slice of the rope directly under the pulley (thin compared with radius of pulley). What does this tell you about the tension each side?

 

[bigu01]

Tension is the same. I tried to find similar triangles and worked with them but I am getting equations with x^4.

 

[Nickie]

I would first start by identifying the key components of this problem, which are the pulley, the cable ABC, and the 100 kg sack. I would also note that the problem specifies neglecting the size of the pulley, which means we can assume it has no mass and does not add any additional forces to the system.

Next, I would draw a free body diagram for the system and label all the forces acting on it. In this case, we have the weight of the sack (980N) acting downwards, the tension in cable AB (TAB) acting upwards and to the right, and the tension in cable BC (TBC) acting upwards and to the left.

Using the given equations, I would then set up two equations for equilibrium: one in the x direction and one in the y direction. In the x direction, we have TABcosθ = TBCCos\phi, and in the y direction, we have TABsinθ + TBCsin\phi = 980N.

To solve for the unknowns, we can use basic trigonometry and the given information about the distance from C to the pulley (3.5-x) to find the values of θ and \phi. Once we have those values, we can plug them into our equations and solve for TAB and TBC.

In conclusion, the key to solving this moving pulley problem is to carefully consider all the forces acting on the system and use basic trigonometry to find the unknowns. It is also important to note any given information, such as neglecting the size of the pulley, that can help simplify the problem.