Analyzing a constraint eqn for a pulley system

In summary, the individual is struggling with creating a constraint equation for a pulley system and is questioning why the last three variables of the constraint are divided by 2. They are then given an explanation that the term represents the distance of the second pulley from ground level and is halved due to the two strings connected to the pulley. Additionally, the system can be analyzed as a mechanism of instantaneous levers, with the anchored pulley acting as a type 1 lever and the movable pulley acting as a type 2 lever with different mechanical advantages.
  • #1
guyvsdcsniper
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Homework Statement
Masses M1 and M2 are connected to a system of strings and pulleys
as shown. The strings are massless and inextensible, and the
pulleys are massless and frictionless. Find the acceleration of M1
Relevant Equations
f=ma
I am having trouble creating the constraint equation for this pulley system.

I don't understand why the last 3 variables of the following constraint is divided by 2?

Could anyone help me understand why this is?

Screen Shot 2022-02-10 at 1.45.07 PM.png
 
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  • #2
quittingthecult said:
I don't understand why the last 3 variables of the following constraint is divided by 2?
That term represents the distance of the second pulley from ground level. (The entire expression is equivalent to saying that the height of the first pulley is fixed.)
 
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  • #3
Doc Al said:
That term represents the distance of the second pulley from ground level. (The entire expression is equivalent to saying that the height of the first pulley is fixed.)
Oh I see. So since there are two strings connected to the second pulley, that distance is halved?
 
  • #4
quittingthecult said:
So since there are two strings connected to the second pulley, that distance is halved?
Right.
 
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  • #5
quittingthecult said:
...
I am having trouble creating the constraint equation for this pulley system.

I don't understand why the last 3 variables of the following constraint is divided by 2?

Could anyone help me understand why this is?
You can also analyze the system as a mechanism of instantaneous levers.
The anchored pulley works as a type 1 lever (mechanical advantage = 1).
The movil pulley works as a type 2 lever (mechanical advantage = 2 for M2 respect to M1, or 0.5 for M1 respect to M2).

1644599743788.png
 
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Related to Analyzing a constraint eqn for a pulley system

1. How do you determine the constraint equation for a pulley system?

To determine the constraint equation for a pulley system, you must first identify all the forces acting on the system and their directions. Then, use the principle of virtual work to find the relationship between the forces and the displacements of the pulleys.

2. What is the purpose of analyzing a constraint equation for a pulley system?

The purpose of analyzing a constraint equation for a pulley system is to understand the mechanical behavior of the system and determine the forces acting on the pulleys. This information is crucial for designing and optimizing the system for maximum efficiency.

3. How do you account for friction in a constraint equation for a pulley system?

Friction can be accounted for in the constraint equation by including a friction coefficient in the calculations. This coefficient represents the resistance of the surfaces in contact and can be determined experimentally or estimated based on the materials and conditions.

4. Can a constraint equation for a pulley system be simplified?

Yes, a constraint equation for a pulley system can be simplified by making certain assumptions, such as ignoring the weight of the pulleys or assuming idealized conditions. However, these simplifications may affect the accuracy of the results.

5. What are some common applications of analyzing constraint equations for pulley systems?

Constraint equations for pulley systems are commonly used in mechanical engineering and design, such as in the construction of elevators, cranes, and other lifting mechanisms. They are also used in physics and mathematics to study the principles of mechanics and motion.

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