Lagrangian with springs and rotating wheels

In summary, a cart with a mass of M is mounted on four frictionless wheels and has a sloped top with an angle alpha. A mass m is suspended from the top of the slope and is able to slide without friction. The Lagrangian for this system is 1/2 (4m'+M+m) x1^2 + 2I (omega)^2 + 1/2m (x1+x2 + 2x1x2cos(alpha)) + mgx2sin(alpha) - 1/2kx2^2. The frequency of oscillation for the mass m' can be found by substituting for x1 and solving for x2 in the equations of motion.
  • #1
mekrob
11
0

Homework Statement


A cart of mass M rides on four frictionlesly mounted wheels of radius a and mass m'. The top of the cart is sloped at an angle alpha horizontally and a mass m is suspended from the top of the slope of force constant k. m slides without friction up or down the slope.

a) Write the Lagrangian.
b) What is the frequency of oscillation of the mass m'.

Homework Equations


L= T-U
[tex]I(for disk/wheel)=1/2MR^2[/tex]

The Attempt at a Solution


[tex]L= 1/2 (4m'+M+m)\dot{x}_{1}^2 + 2I (\omega)^2 + 1/2m(\dot{x_{1}}+\dot{x_{2}} + 2\dot{x_{1}} \dot{x_{2}}cos(\alpha)) + mgx_{2}sin(\alpha) - 1/2kx_{2}^2[/tex]

x1 = position of M
x2 = position of m
 
Last edited:
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  • #3
What is your question? If you want someone to verify that equation for you, it'd help if you explained how you got it.
 
  • #4
What is your question?
a) Write the Lagrangian.
This was on an exam (studying for the final). I wrote the same answer on the exam and the only mark the prof made was asking which are [tex]x_{1}[/tex] and[tex] x_{2}[/tex], which I had forgot to clarify. I received most of the points for the question, so I assume it's right.

b) What is the frequency of oscillation of the mass m'.
 
  • #5
We don't do problems for you. Show what you've tried and we'll try to push you in the right direction. Start by writing the equations of motions, and try to figure out how you would extract a frequency from them.
 
  • #6
Going through the Lagrangians, I got
[tex]\ddot {x_{1}} = \frac {gsin(\alpha) - \frac {kx_{2}}{m} - \ddot{x{_2}}}{cos(\alpha)}[/tex]

and

[tex](4m' + m + M) \ddot{x_{1}} + 2I\omega^2 + m(\ddot{x_{1}} + \ddot{x_{2}}cos\alpha) = constant [/tex]

Again, I got these right on the exam, apparantely, I was just never good at converting this to [itex]\omega[/itex]. Any help getting started?
 
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  • #7
You need to write [itex]\omega[/itex] (ie, the omega for the rotating wheels, not the one you're trying to find for the oscillation of m) in terms of [itex]x_1[/itex], which will change your second equation. Then this will tell you that [itex]\ddot{x_1}[/itex] is proportional to [itex]\ddot{x_2}[/itex], so you can substitute for [itex]\ddot{x_1}[/itex] in the first equation to get an ODE for [itex]x_2[/itex] which should be easy to solve.
 

What is a Lagrangian with springs and rotating wheels?

A Lagrangian with springs and rotating wheels is a mathematical model used to describe the motion of a system that includes both springs and rotating wheels. It takes into account the kinetic and potential energies of the system to determine the equations of motion.

How is a Lagrangian with springs and rotating wheels different from other mathematical models?

A Lagrangian with springs and rotating wheels is different from other mathematical models because it takes into account both rotational and linear motion, as well as the energy stored in the springs. This allows for a more accurate description of the system's motion.

What are the equations used in a Lagrangian with springs and rotating wheels?

The equations used in a Lagrangian with springs and rotating wheels include the Lagrangian function, which is a combination of the kinetic and potential energies of the system, and the Euler-Lagrange equations, which are used to determine the equations of motion.

What are the applications of a Lagrangian with springs and rotating wheels?

A Lagrangian with springs and rotating wheels is commonly used in the field of mechanics and engineering to model and analyze systems such as mechanical systems, robotic arms, and vehicle suspensions.

What are the limitations of using a Lagrangian with springs and rotating wheels?

One limitation of using a Lagrangian with springs and rotating wheels is that it assumes the system is in a state of equilibrium and does not take into account external forces or disturbances. It also does not account for non-linearities in the system.

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