Solving for the motion of a 2 mass system using Lagrangian mech.

• Wavefunction
In summary, in this conversation, the participants discuss a homework problem involving two equal masses constrained by a spring-and-pulley system. The goal is to derive the equations of motion using Lagrangian methods and solve for the extension of the spring as a function of time. The conversation also includes discussions on constraints and boundary conditions. The final solution involves a simple harmonic oscillator with a constant driving force. One participant also asks for a sketch of the system.
Wavefunction

Homework Statement

Two equal masses are constrained by the spring-and-pulley system shown in the accompanying
sketch. Assume a massless pulley and a frictionless surface. Let $x$ be the extension of the
spring from its relaxed length. Derive the equations of motion by Lagrangian methods. Solve
for x as a function of time with the boundary conditions $x = 0$,$\frac{dx}{dt} = 0$ at $t =0$

Homework Equations

(1)$\frac{∂L}{∂q_j}-\frac{d}{dt}\frac{∂L}{∂\dot{q_j}}+\sum_k λ_k(t)\frac{∂f_k}{∂q_j} = 0$

The Attempt at a Solution

First I'll construct the Lagrangian $L=T-U$ , $T= \frac{m}{2}[\dot{x}^2+\dot{y}^2]$ , $U = \frac{kx}{2}+mgy → L = \frac{m}{2}[\dot{x}^2+\dot{y}^2] - \frac{kx}{2}-mgy$

Okay now, I also have the constraint that the total length of the string doesn't change. Also when I shift the x-coordinate in the positive direction the shift in the y-coordinate will be negative which implies:

$dx=-dy → x = -y+C$ and let $C=0$ so that $x+y = 0$

Now, I have my constraint $f(x,y) = 0$ so now I can use (1) in order to find the equations of motion for the system:

(2) $\frac{∂L}{∂x} = -kx$, $\frac{d}{dt}\frac{∂L}{∂\dot{x}} = \frac{d}{dt}[m\dot{x}] = m\ddot{x}$, and $λ\frac{∂f}{∂x} = λ$

(3) $\frac{∂L}{∂y} = -mg$, $\frac{d}{dt}\frac{∂L}{∂\dot{y}} = \frac{d}{dt}[m\dot{y}] = m\ddot{y}$, and $λ\frac{∂f}{∂y} = λ$

Now, I simply add the corresponding pieces together:

(2') $-kx-m\ddot{x}+λ = 0$

(3') $-mg-m\ddot{y}+λ = 0$

From my constraint I can replace $\ddot{y}$ with $-\ddot{x}$ and add the two equations to solve for $λ$ doing so gives:

(2')+(3'): $-kx-mg+2λ = 0 → λ = \frac{kx+mg}{2}$

Now that I have $λ$ I can plug it into either (2') or (3') to solve for $x$ I'll choose (2'):

$-\frac{kx}{2} + \frac{mg}{2} - m\ddot{x} = 0 → \ddot{x}+\frac{k}{2m}x = \frac{g}{2}$

I'll define $ω\equiv \sqrt{\frac{k}{2m}}$ so that now the above DE is of the form of a simple harmonic oscillator with a constant driving force:

$x(t)= x_c(t) + x_p(t)$, $x_c(t) = Acos(ωt)+Bsin(ωt)$, and $x_p(t) = C$

$\ddot{x_p}+ω^2x_p = \frac{g}{2} → C=\frac{g}{2ω^2} = \frac{gm}{k}$

Then $x(t) = Acos(ωt)+Bsin(ωt)+ \frac{gm}{k}$

Now apply the boundary conditions:

$0 = A+\frac{gm}{k}$

$\dot{x} = -Aωsin(ωt)+Bωcos(ωt) → 0=Bω → x(t)=-\frac{gm}{k}cos(ωt)+\frac{gm}{k}$

Finally, $x(t) = \frac{gm}{k}[1-cos(\sqrt{\frac{k}{2m}}t)]$ and of course $y(t) = -x(t)$ so
$y(t) = -\frac{gm}{k}[1-cos(\sqrt{\frac{k}{2m}}t)]$

So I'm fairly certain I got this correct (but it never hurts to check!) I'm fairly certain because my forces of constraint are the same for both coordinates (i.e tension in the string is the same everywhere) Also I ended up having only one canonical coordinate which corresponds to one dof : $s=Dn-m$ where $s$ is the number of degrees of freedom, $D$ is the number of dimensions each particle is allowed to move in, $n$ is the number of particles, and $m$ is the number of constraints on the coordinate(s). Using this relation: $s=(1)(2)-(1)=1$ which is exactly the number of coordinates I ended up with at the end. Thank you in advance for looking over my work. (:

Where's the sketch?

I have attached the image in this message.

Attachments

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1. What is a 2 mass system?

A 2 mass system refers to a physical system that consists of two masses connected by a spring or other types of forces. This system is commonly used to model many real-world situations, such as a pendulum or a double-spring system.

2. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework for analyzing the motion of a system by using the Lagrangian function, which takes into account the kinetic and potential energies of the system. It is an alternative to Newtonian mechanics and is often used to solve complex systems with multiple degrees of freedom.

3. How do you solve for the motion of a 2 mass system using Lagrangian mechanics?

The first step is to define the Lagrangian function for the system, which is the difference between the kinetic and potential energies. Then, you can use the Euler-Lagrange equations to find the equations of motion for each mass in the system. Finally, you can solve these equations to determine the position, velocity, and acceleration of each mass over time.

4. What are the advantages of using Lagrangian mechanics over Newtonian mechanics?

There are several advantages of using Lagrangian mechanics, including its ability to handle complex systems with multiple degrees of freedom, its ability to incorporate constraints and generalized coordinates, and its elegant mathematical formulation. It also provides a more systematic approach to solving problems compared to the trial-and-error method used in Newtonian mechanics.

5. What are some real-world applications of solving for the motion of a 2 mass system using Lagrangian mechanics?

Lagrangian mechanics has numerous applications in physics and engineering, such as in the study of celestial mechanics, robotics, and fluid dynamics. It is also used in the design and analysis of mechanical systems, such as suspension systems in cars and aircraft landing gear.

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