# Find the equation of motion using the Lagrangian for this Atwood machine

• Istiak
In summary, the tension in the two ropes can be different, and the machine providing a mechanical advantage of 2 also changes the tension in the ropes.
Istiak
Homework Statement
Find equation of motion using Lagrangian of Atwood machine
Relevant Equations
L=T-U
##\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}})=\frac{\partial L}{\partial q}##
My understanding of the system from the image (which was given in book)

I could see there's 3 tension in 2 body. Even I had seen 2 tension in a body. It was little bit confusing to me. I could find tension in Lagrangian from right side. But left side was confusing to me.

$$L=\frac{1}{2}m_1\dot{x}^2+\frac{1}{2}m_2\dot{x}^2-m_1gx-m_2g(l-x)-m_2g(l_1-x)$$
Here $l-x$ is representing the potential energy for "center rope" tension and l_1-x is representing tension for right one.

After using Euler form and rearranging I get that

$$\ddot{x}=\frac{2m_2-m_1}{m_1+m_2}g$$

I don't know if the answer is correct. I know that the acceleration is for whole body. But in the book, they had found separated acceleration

From their equation it's like actually my answer is wrong.

Hi,

Your expose is rather confusing to me. You don't say what ##x## is. ##l## ? ##l_1## ?

Are you suggesting ##m_1## and ##m_2## both move with velocity ##\dot x## ?

##\ ##

Yes! I am assuming that both object moves at same speed which I denoted by ##dot x##. And l and l_1 is constant length. l is from left to center (where object m_2 is connected. And l_1 is right side length. I was thinking there's two tension hence they will act different way hence had taken separated length.

When first object is moving then it will pull the rope either (earlier i was thinking that the object is attached with the rope but its not hence my assumption was wrong.)

My mind was stucked with two tension and two body. But the system was messy by additional tension which i was unable to understand hence i am thinking that my idea behind ##x## was also wrong.

Why not make two drawings, with m1 in two different postions, and measure !

##\ ##

Why are you assuming ##m_1## and ##m_2## share the same velocity?

Direct an axis ##z## downwards vertically and let ##z_1,z_2## be coordinates of the particles. Find a relation between ##z_1,z_2##. Use the length of the string. Differentiate the relation in time

I was trying to do the solution another way. What still disturbing me that is third no. tension. I was thinking if I should consider another kinetic energy in terms of third tension. In my new step, I was assuming that there's another K.E. in terms third tension.

$$L=\frac{1}{2}m_1\dot{x}^2+\frac{1}{2}m_2\dot{x}^2+\frac{1}{2}m_2\dot{x}^2-m_2gx-m_2g(l_b-l_a-x)-m_2g(l_a-x)$$
After using Euler-Lagrange,
$$\ddot{x}=\frac{2m_2-m_1}{m_1+2m_2}g$$

Is my answer correct now? I think it's not correct cause, there won't be 3 K.E for only 2 body. There must be 2 K.E. for 2 body we just have to rearrange them. So I think the accurate answer is
$$\ddot{x}=\frac{2m_2-m_1}{m_1+m_2}g$$

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1) There is no way for the tension of the rope to magically change between the point of anchorage to the ceiling and the opposite end.

2) During the same time, body of mass m2 moves distance d and body of mass m1 moves 2d; therefore, bodies 1 and 2 can't have same average velocities or instantaneous accelerations.

3) The function of the fixed-to-ceiling pulley is only to change the direction of the rope and its associate tension, but the function of the moving pulley is more complicated: it is also a simple machine providing a mechanical advantage of value 2.

4) Note the values to which a1 and a2 tend to when the values of m2 and m1 tend to zero in the solution equations shown in the book. What rate of m1/m2 would stop the system from acelerating in any direction?

http://hyperphysics.phy-astr.gsu.edu/hbase/atwd.html#c3

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## 1. What is an Atwood machine?

An Atwood machine is a simple mechanical device used to demonstrate the principles of classical mechanics. It consists of two masses connected by a string or cable that passes over a pulley. The machine is named after George Atwood, who first described it in the late 18th century.

## 2. What is the Lagrangian method?

The Lagrangian method is a mathematical approach used to describe the motion of a system by considering its potential and kinetic energies. It is based on the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action integral. In the case of an Atwood machine, the Lagrangian method can be used to find the equation of motion.

## 3. How do you find the equation of motion using the Lagrangian for an Atwood machine?

To find the equation of motion using the Lagrangian for an Atwood machine, we first need to determine the Lagrangian of the system. This involves calculating the potential and kinetic energies of the masses and the tension in the string. Once we have the Lagrangian, we can use the Euler-Lagrange equations to find the equations of motion for each mass.

## 4. What are the assumptions made when using the Lagrangian method for an Atwood machine?

When using the Lagrangian method for an Atwood machine, we make the following assumptions: 1) the string is massless and does not stretch, 2) the pulley is frictionless, 3) the masses are point masses, and 4) the only force acting on the system is gravity. These assumptions allow us to simplify the equations and focus on the essential principles of the system.

## 5. How is the equation of motion for an Atwood machine used in real-world applications?

The equation of motion for an Atwood machine can be used in various real-world applications, such as in elevator systems, cranes, and other lifting mechanisms. It can also be used to study the motion of objects in a gravitational field and to understand the principles of energy conservation and mechanical work. Additionally, the Lagrangian method can be applied to more complex systems, making it a valuable tool in many areas of science and engineering.

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