Equations of Motion Homework: Lagrange & Newton's 2nd Law

In summary, the conversation discusses a problem involving the Lagrange method and Newton's second law. The attempt at a solution includes equations for mass Mo and a rolling mass m, but there are difficulties matching the remaining equations. The solution does not match when organizing everything into the x-y or x'-y' plane, and the question arises of whether the momentum of the rolling mass should be included with the external forces acting on mass M.
  • #1
rcummings89
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Homework Statement


Please see the attached picture for the problem description. Now, I have a solution using the Lagrange method, (it should coincide with Newton's second law, I believe?) I just have a hard time getting my equations of motion to match.

Homework Equations


∑Fext - M dv/dt = 0


The Attempt at a Solution


For mass Mo I have
K(x2 - x1) = Mo*x1'' x (where mass Mo moves a distance x1 and mass M moves a distance x2) which agrees with the Lagrange solution

I have trouble matching the remaining equations. For the rolling mass m, I have the external forces as -mg (y-dir) and FN (y'-dir) where x' and y' are at an angle θ depending on the location of the rolling mass.

Then my equation of motion is -mg y + N y' = m[x'' x + (R-r)θ'' y' - (R-r)θ'2 x']

Now I realize that I need to organize everything into the x-y or x'-y' plane but when I do the solution doesn't match; do you see anything that I'm missing?

Finally for mass M

∑Fext = -mg y - K(x2 - x1) x
My professor typically neglects gravity and normal forces acting on carts, so I don't include them here. But I feel like I'm neglecting a force. Should I include the momentum of the rolling mass with the external forces acting on M? Because setting what I have = M*x2'' x doesn't match, and I feel like I'm over-simplifying the problem.

Thanks in advance
 

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FAQ: Equations of Motion Homework: Lagrange & Newton's 2nd Law

1. What is the difference between Lagrange's and Newton's Second Law of Motion?

Lagrange's Second Law of Motion, also known as the Euler-Lagrange equation, is a generalized form of Newton's Second Law that can be used to describe the motion of a system with any number of particles and any number of constraints. It takes into account the total potential and kinetic energy of the system, while Newton's Second Law only considers the forces acting on a single point particle.

2. How do you write the equations of motion using Lagrange's and Newton's Second Law?

Lagrange's Second Law can be written as:
∂L/∂q - d/dt(∂L/∂q') = 0, where L is the Lagrangian (total kinetic energy minus the total potential energy) and q represents the generalized coordinates of the system.
Newton's Second Law can be written as:
F = ma, where F is the sum of all the external forces acting on the system, m is the mass of the object, and a is the acceleration of the object.

3. What are the advantages of using Lagrange's Second Law over Newton's Second Law?

One advantage of using Lagrange's Second Law is that it takes into account all the forces and energies acting on a system, rather than just the external forces on a single point particle. This makes it more applicable to real-world systems with multiple particles and constraints. Additionally, Lagrange's Second Law is often simpler and easier to solve than Newton's Second Law for more complex systems.

4. How do you use Lagrange's Second Law to find the equations of motion for a system?

To use Lagrange's Second Law, you first need to determine the Lagrangian of the system, which is the total kinetic energy minus the total potential energy. Then, you take the partial derivative of the Lagrangian with respect to each generalized coordinate. Finally, you take the derivative of the partial derivative with respect to time and set it equal to zero. This will give you a set of equations that can be solved to find the equations of motion for the system.

5. Can Lagrange's Second Law be used for systems with non-conservative forces?

Yes, Lagrange's Second Law can be used for systems with non-conservative forces, such as friction or air resistance. The Lagrangian can be modified to include these non-conservative forces, and the resulting equations of motion will take these forces into account. This makes Lagrange's Second Law more versatile than Newton's Second Law, which cannot account for non-conservative forces.

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