1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Constaint to a rolling object

  1. Jan 4, 2009 #1


    User Avatar

    1. The problem statement, all variables and given/known data
    A cylinder, radius R, mass M and moment of inertia I, is rolling on a horizontal surface without slipping and a constant force F is exerted on the center of the cylinder horizontally to the right. Try to find the friction by solving the Lagrangian equation.

    2. The attempt at a solution
    Since the object is rolling without slip, there must be a friction force exerted at the contact point to the left.

    When we go to the Lagrangian mechanics, let's call the center of the cylinder (x, y) and the angle made by a specific point on the edge of the cylinder to the vertical line passing through center [tex]\theta[/tex]. Since the object is moving on the horizontal level, no potential. The constraint for no-slipping action is

    [tex]f = R\theta - x =0[/tex]

    The Lagrangian is

    [tex] L = \frac{M}{2}\dot{x}^2 + \frac{I}{2}\dot{\theta}^2[/tex]

    The Lagrangian equation is

    \frac{d}{dt}\frac{\partial L}{\partial\dot{x}} = \lambda \frac{\partial f}{\partial x} = -\lambda


    \frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}} = \lambda \frac{\partial f}{\partial \theta}
    = R\lambda

    where [tex]\lambda[/tex] is the Lagrange undetermined multiplier. From these equations, I get [tex]\lambda=F[/tex], but how do I relate undetermined multiplier to friction?

    If we start from newton's law, it is very easy to solve for the friction, which is obvious written in terms of F. But from my solution above, seems cannot get the same form. What's going on with my solution?
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?

Similar Discussions: Constaint to a rolling object
  1. Object dynamics (Replies: 0)