Why Does a2 Equal m1g/m2 in an Accelerating Pulley System?

• CausticPhantom
In summary, the conversation discusses the equation for calculating the force and acceleration of a system consisting of masses m1, m2, and M. The correct equation is F = (m1g)(M+m1+m2)/(m2), as the force m1g only acts on m2, not the entire system. The incorrect conclusion of m1g/(m1+m2) as the acceleration of the system is due to not considering all the forces acting on the individual masses.
CausticPhantom
Member advised to use the homework template for posts in the homework sections of PF.

a1 = a2
a1 = F/(M+m1+m2) | Force/(Mass of entire system) = acceleration of entire system
*a2 = (m1g)/(m2+m1) | Force = m1g; acceleration of m2 and m1 = m1g/(m1+m2)

My answer: F = (m1g)(M+m1+m2)/(m2+m1)
The book's answer: F = (m1g)(M+m1+m2)/(m2)
*This step is what leads me to a slight variation of the book's provided answer.

I've looked through the forums, and have done a lot of thinking myself, and I believe what it comes down to is a false assumption. I understand that the force pulling m1 and m2 is m1g, and that the tension that then pulls m2 is equal to m1g, leading to an acceleration of m1g/m2 and subsequently the correct answer, but I do not understand what is incorrect about concluding that the system (looking at m2 and m1) as a whole accelerates at m1g/(m1+m2). Gut feeling tells me that the conclusion I've made is incorrect because the acceleration would be m1g/(m1+m2) in an inertial frame, but not in this case because it is part of an accelerating system.

Why does a2 = m1g/m2 rather than a2 = m1g/(m1+m2)?

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CausticPhantom said:
but I do not understand what is incorrect about concluding that the system (looking at m2 and m1) as a whole accelerates at m1g/(m1+m2).
You have here considered m1 and m2 as a system, but the force you quote is not the force on that system, it is the force on m2 only.

Orodruin said:
You have here considered m1 and m2 as a system, but the force you quote is not the force on that system, it is the force on m2 only.

Is there another force that I would need to account for, such as the applied force on M?

CausticPhantom said:
Is there another force that I would need to account for, such as the applied force on M?
There are forces acting from the wall on m1 and from the pulley on the rope. Computing these forces is not necessary if you chose a better system to consider.

What is an accelerating pulley system?

An accelerating pulley system is a mechanical system that consists of two or more pulleys connected by a belt or rope. It is designed to transfer force and motion from one object to another, typically with a mechanical advantage.

How does an accelerating pulley system work?

An accelerating pulley system works by using the principle of mechanical advantage, where the force applied to one end of the rope is multiplied and applied to the other end. As the rope passes over each pulley, the force is multiplied, allowing for heavier objects to be lifted with less force.

What are some real-world applications of an accelerating pulley system?

Accelerating pulley systems are used in a variety of applications, such as elevators, cranes, and conveyor belts. They are also commonly used in exercise equipment, such as weight lifting machines, to provide resistance and increase the effectiveness of workouts.

What factors affect the efficiency of an accelerating pulley system?

The efficiency of an accelerating pulley system can be affected by several factors, including the number of pulleys, the angle of the rope, and the friction between the rope and pulleys. In general, the more pulleys in the system, the higher the mechanical advantage and the more efficient the system will be.

Are there any limitations to an accelerating pulley system?

While accelerating pulley systems can provide mechanical advantage and increase efficiency, they are limited by the laws of physics. The weight of the rope and the friction between the rope and pulleys can decrease the overall efficiency of the system. Additionally, there is a limit to how much weight can be lifted using a specific number of pulleys.

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