How Do You Solve a Non-Equilibrium Pulley System with a 154 Degree Angle?

In summary, to solve a non-equilibrium pulley system with a 154-degree angle, one must analyze the forces acting on the system, including gravitational forces, tension in the ropes, and the angles involved. Using trigonometric functions, the components of the forces can be resolved to establish equilibrium conditions. Apply Newton's second law to set up equations that account for all forces and their angles, then solve the resulting system of equations to find the unknown tensions and accelerations. Careful consideration of the geometry and the relationships between the forces is essential for accurate solutions.
  • #1
ashley2024
5
0
Homework Statement
Find the tension in the following system (see image).
Relevant Equations
Fnet = ma, Ff = uFn,
Please note that the system is not in equilibrium, and that tension must be solved for the instant where the angle is 154 degrees.

inst V.png


My attempt (correct ans is Ft = 626N)
image (4).png
 
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  • #2
##a_1=a_2##? Sure?
 
  • #3
haruspex said:
##a_1=a_2##? Sure?
Nope, but I don't really have any other ideas.
 
  • #4
ashley2024 said:
Nope, but I don't really have any other ideas.
There is a simple relationship between the two, just not quite that simple.
Start with positions. If m is x from the vertical through M, the knot is y below the horizontal through m, the string connecting them has length L, and it is at θ to the vertical, write expressions for x and y and differentiate twice.
 
  • #5
haruspex said:
There is a simple relationship between the two, just not quite that simple.
Start with positions. If m is x from the vertical through M, the knot is y below the horizontal through m, the string connecting them has length L, and it is at θ to the vertical, write expressions for x and y and differentiate twice.
Would this be similar to the related rates problems? Since the ropes have fixed length, it seems similar to the ladder problem where you have to differentiate its rate as it falls...
 
  • #6
ashley2024 said:
Please note that the system is not in equilibrium...
Welcome, Ashley! :smile:

What makes you state that?
 
  • #7
Lnewqban said:
Welcome, Ashley! :smile:

What makes you state that?
The teacher who gave this problem.
 
  • #8
ashley2024 said:
The teacher who gave this problem.
Then, the tension in the V=shaped wire should be the minimum to keep masses #1 sliding toward each other.
We could then, disregard the value of the force exerted by the falling mass #2, since we know that it is plenty to have induced and to keep the sliding movements of both masses #1.
Therefore, calculating T2 seems not to be necessary.
As the system is geometrically symmetrical, there is a unique value for T1.
 
  • #9
Interesting! But I don't understand why it would be the minimum though? Since its asking for the tension when the angle is 154, I assumed that the tension could be still in the process of changing.
 
  • #10
ashley2024 said:
But I don't understand why it would be the minimum though
It isn't. Don’t be distracted by that.
 
  • #11
ashley2024 said:
Would this be similar to the related rates problems? Since the ropes have fixed length, it seems similar to the ladder problem where you have to differentiate its rate as it falls...
Yes.
 

FAQ: How Do You Solve a Non-Equilibrium Pulley System with a 154 Degree Angle?

What is a non-equilibrium pulley system?

A non-equilibrium pulley system is a mechanical setup where the forces acting on the system are not balanced, resulting in acceleration. This contrasts with an equilibrium system where the sum of forces and torques equals zero, leading to no acceleration.

How do you account for the 154-degree angle in the pulley system?

To account for the 154-degree angle, you need to resolve the forces acting on the system into their components. Typically, this involves breaking down the tension forces in the ropes into horizontal and vertical components using trigonometric functions like sine and cosine.

What equations are used to solve the forces in a non-equilibrium pulley system?

The primary equations used are Newton's second law (F = ma) for each mass in the system. Additionally, you will use trigonometric identities to resolve forces and the constraints imposed by the pulleys and ropes. Summing the forces in the horizontal and vertical directions will give you a set of simultaneous equations to solve.

How do you determine the acceleration of the masses in the system?

To determine the acceleration, you need to set up the equations of motion for each mass based on the forces acting on them. Once you have the equations, you solve them simultaneously to find the acceleration. This typically involves using the net force and the mass of each object to find the acceleration using F = ma.

What role does friction play in solving a non-equilibrium pulley system?

Friction can significantly affect the forces in a pulley system. If friction is present, it must be included in the force calculations. This typically involves adding frictional force components, which are usually proportional to the normal force and depend on the coefficient of friction between the surfaces in contact.

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