3 weights suspended on 2 pulleys to form a M shape

[lksmith17]

Homework Statement

A block with a mass of 25 kg is hung midway between two pulleys, with a rope connecting a hook at the top of the block to each of the two pulleys. On the other side of the pulleys, both ropes are connected to blocks with masses of 15 kg. The 25 kg mass in the middle sags downward until the 3 blocks and 2 pulleys form the shape of an “M”. If the pulleys are 45 cm apart, how far below the level of the pulleys does the top of the 25 kg mass need to be in order for the system to be in equilibrium?


2. Homework Equations

Sum of Forces =0

3. The Attempt at a Solution
I have calculated the tensions in one rope to be 147N using 15kg*9.8m/sec^2 and the Fg of the middle block is 245N I know I need to find the angles of the rope to the ceiling by splitting the tension of the rope in x and y directions, but I can not figure out how to get those values.

 

[PhanthomJay]

You know the tension in the rope, and you know the weight of the block in the middle. At the joint where they all meet, identify the forces acting in the y direction and use Newton 1. Then it's geometry.

 

[lksmith17]

At the joint where they all meet is at the middle block right? If so, the Force in the y direction 245N down from gravity but the tension is is from the two ropes is the force that counters the force of gravity. So I can make a right triangle, but i only have the hypotenuses' length. I don't have any other angles to figure out the the angle between the ceiling and the roof. I guess I am missing something completely could you let me know what I am doing wrong. Am I not identifying a force correctly?

 

[PhanthomJay]

From Newtons 1st law in the y direction, the sum of the vertical components of the tension forces in the ropes must equal the weight of the center block. So what is the sum of those tension components, and what is the vert component of the tension in one of the ropes. The tension force acts along the diagonal, so use Pythagoras to get the horiz comp. then do some geometry and trig to find the requested distance.

 

[Nickie]

I would approach this problem by first drawing a free body diagram of the system, labeling all the forces acting on each object (blocks and pulleys) and using the equations of motion to analyze the system. From the given information, we can determine that the system is in equilibrium, meaning that the sum of all forces acting on each object must be equal to 0.

To find the distance below the level of the pulleys where the top of the 25 kg mass needs to be, we can use the equation of torque, which states that the net torque acting on an object must be equal to 0 for it to be in rotational equilibrium. We can also use the equation of torque to determine the angles of the ropes to the ceiling.

First, we can calculate the tension in each rope using the equation F=ma, where F is the tension, m is the mass, and a is the acceleration due to gravity. We can then split this tension force into its vertical and horizontal components, using basic trigonometry. The vertical component of the tension force will be equal to the weight of each block, while the horizontal component will be equal to the tension in the rope.

Next, we can calculate the net torque acting on each block by multiplying the force acting on each block by the distance from the pivot point (in this case, the pulley). The torque due to the weight of each block will be equal to the weight multiplied by the distance from the pivot point, while the torque due to the tension force will be equal to the tension force multiplied by the distance from the pivot point.

We can then set up an equation for the net torque acting on each block, and solve for the distance at which the top of the 25 kg mass needs to be in order for the system to be in equilibrium.

In conclusion, as a scientist, I would approach this problem by using the equations of motion and torque to analyze the system and determine the distance at which the top of the 25 kg mass needs to be in order for the system to be in equilibrium.