Two masses connected by Pulley - Lagrangian problem

In summary, the problem of two masses connected by a pulley involves analyzing the dynamics of a system where two weights are suspended on either side of a frictionless pulley. The Lagrangian approach is employed, which involves defining the kinetic and potential energies of the masses and deriving the equations of motion using the principle of least action. The system is characterized by its constraints and the interactions between the masses, allowing for the determination of their motion over time. This method provides insights into the conservation of energy and the forces at play in the system.
  • #1
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1718493064484.png

My solution to (a) is,
We have constraint . There are many places we could define our (x,y) Cartesian coordinate system. However, the most easiest I think for the problem would be to attach a and coordinate system at the COM of . We define parallel to rightward horizontal and parallel to upwards vertical.
1718493234131.png

Since the COM of is not moving with respect to our defined coordinate system (orange dot denotes m_1 COM in diagram above), then we omit KE in the Lagrangian form of the system. The coordinates of is

The velocity coordinates are by definition the time with respect to the linear time (I think we could generalize this from being with respect to linear time to being with respect to any non-linear, higher dimensional time ). However, we assume classical case. Thus

Thus by definition of T and V, Lagrangian is

However, does anybody please know whether I am correct so far?

Thanks!
 
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  • #2
At a glance your Lagrangian does not have m1. Dimension of potential energy term has excess T^-1. Are they OK?
 
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  • #3
anuttarasammyak said:
At a glance your Lagrangian does not have m1. Dimension of potential energy term has excess T^-1. Are they OK?
Thank you for your reply @anuttarasammyak !

The Lagrangian does not have m1 since I chose m1 COM as coordinate system. Not sure what you mean about dimension of PE term sorry.

Thanks!
 
  • #4
ChiralSuperfields said:
The Lagrangian does not have m1 since I chose m1 COM as coordinate system.
I don't understand this, please explain. Are you saying that the equations of motion will be the same regardless of the size of ? If is that of a battleship, the EOM of will be pretty close to that of a pendulum. If is that of a gnat, the EOM of will be be pretty close to that of a free-falling object..
 
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  • #5
ChiralSuperfields said:
Not sure what you mean about dimension of PE term sorry.

the second term, containing g, has physical dimension of MLT^-2LT^-1=(ML^2T^-2)T^-1 but it should have dimension of ML^2T^-2, energy.
 
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  • #6
anuttarasammyak said:

the second term containing g has physical dimension of MLT^-2LT^-1=(ML^2T^-2)T^-1 but it should have dimension of ML^2T^-2, energy.
The first term also suffers from the condition of the same bad dimensions.
 
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I also have concerns about dimensions)

Your second term has in it along with generalized velocities. It seems to me that you’re mixing your Kinetic and Potential terms.

Why would a potential term have generalized velocities? Also, why would kinetic terms have acceleration ?
 
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FAQ: Two masses connected by Pulley - Lagrangian problem

What is the Lagrangian formulation of the two masses connected by a pulley system?

The Lagrangian formulation involves defining the Lagrangian function, which is the difference between the kinetic energy (T) and potential energy (V) of the system. For two masses \( m_1 \) and \( m_2 \) connected by a pulley, the Lagrangian \( L \) can be expressed as \( L = T - V \). The kinetic energy is the sum of the kinetic energies of both masses, and the potential energy is determined by their positions relative to a reference point, typically the height of the masses.

How do you derive the equations of motion using the Lagrangian?

To derive the equations of motion, you apply the Euler-Lagrange equation, which states that \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 \), where \( q \) represents the generalized coordinates of the system. For the two masses, you typically choose the displacement of one mass as the generalized coordinate and express the other mass's displacement in terms of it. By substituting these into the Euler-Lagrange equation, you can derive the equations of motion for the system.

What are the constraints in a two-mass pulley system?

The primary constraint in a two-mass pulley system is the inextensibility of the connecting string and the geometry of the pulley. This means that the total length of the string remains constant, which leads to a relationship between the displacements of the two masses. If one mass moves down by a certain distance, the other mass must move up by the same distance, maintaining the constraint of the system.

How does one account for friction in the pulley system when using Lagrangian mechanics?

To account for friction in the pulley system, you can introduce a non-conservative force into the system. This can be done by modifying the potential energy to include a term that represents the work done against friction. Alternatively, you can include a damping term in the equations of motion derived from the Lagrangian, which accounts for energy lost due to friction. This requires adjusting the equations accordingly to reflect the dissipative forces acting on the system.

What are the practical applications of studying a two-mass pulley system using Lagrangian mechanics?

Studying a two-mass pulley system using Lagrangian mechanics has various practical applications, including understanding mechanical systems in engineering, designing elevators, analyzing cranes, and studying dynamics in robotics. The Lagrangian approach provides insights into energy conservation and allows for the analysis of complex systems with multiple degrees of freedom, making it a valuable tool in both

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