Using Lagrangian Formulation to Solve a Mass on a Wedge Problem

[bit188]

Homework Statement

A wedge of mass M and angle [alpha] slides freely on a horizontal plane. A particle of mass m moves freely on the wedge. Determine the motion of the particle as well as that of the wedge.

Note: We are supposed to use the Lagrangian formulation of mechanics to solve this problem.

The Attempt at a Solution

Okay, here's what I've got so far:

We can write the position of the mass m as

x = l cos [alpha];
y = l sin [alpha]

where l is the length of the hypotenuse of the wedge that the mass is sliding on.

Thus, the kintetic energy of the mass m is equal to [(1/2)m][l^2 cos^2(alpha) - l^2 sin^2(alpha)]; the potential energy is equal to [l sin(alpha)][mg]. Therefore, the Lagrangian can be written as

L = [(1/2)l^2m(cos^2(alpha) - sin^2(alpha))] - [l sin(alpha)][mg].

And that's as far as I've gotten; I'm having trouble with the rest. Help is appreciated greatly.

 

[Avodyne]

You need to consider the motion of the wedge. There are two independent degrees of freedom here: the horizontal position of the wedge, and the height of the mass on the wedge. Everything else (such as the horizontal position of the mass) can be expressed in terms of these two.

 

[bit188]

You need to consider the motion of the wedge. There are two independent degrees of freedom here: the horizontal position of the wedge, and the height of the mass on the wedge. Everything else (such as the horizontal position of the mass) can be expressed in terms of these two.

How do I specify the position of the wedge? It's moving freely on a horizontal plane; it doesn't seem like there's a way to tell exactly where it is (at least to me).