Block on a Cylinder, using Lagrange's Equation

[Oijl]

Homework Statement

A hard rubber cylinder of radius r is held fixed with its axis horizontal, and a wooden cube of mass m and side 2b is balanced on top of the cylinder, with its center vertically above the cylinder's axis and four of its sides parallel to the axis.

Assuming that b < r, use the Lagrangian approach to find the angular frequency of small oscillations about the top.

Homework Equations

T (kinetic energy) = (1/2)(mv^2 + I[tex]\dot{\theta}[/tex][tex]^{2}[/tex])
I (moment of inertia about the center of mass) = (2mb^2)/3
U (potential energy)= mg[(r + b)cos[tex]\theta[/tex] + r[tex]\theta[/tex]sin[tex]\theta[/tex]]

The Attempt at a Solution

Now, what would be nice would be to write the coordinates of the center of mass. I can differentiate that and get v, which I plug into T and then I have L = T - U and I can do the problem.

But how can I write down the coordinates of the CM? I never was very good at center of mass problems.

Thanks ahead of time.

 

[anathema]

Thank you for your question. I would approach this problem by first defining the coordinate system. In this case, we can use the horizontal axis passing through the center of the cylinder and the vertical axis passing through the center of mass of the cube. This way, we can define the position of the center of mass of the cube as (0, b) and the position of the cylinder as (r, 0).

Next, we can use the equations for the center of mass to find the coordinates of the center of mass of the combined system. Since the cylinder is held fixed, its position does not change and we can ignore it in our calculations. The center of mass of the combined system can be found using the following equation:

x_cm = (m*x_cube + M*x_cylinder)/(m+M)

where x_cube and x_cylinder are the x coordinates of the center of mass of the cube and cylinder, respectively, and m and M are the masses of the cube and cylinder, respectively.

Similarly, we can find the y coordinate of the center of mass using the equation:

y_cm = (m*y_cube + M*y_cylinder)/(m+M)

where y_cube and y_cylinder are the y coordinates of the center of mass of the cube and cylinder, respectively.

Now that we have the coordinates of the center of mass, we can use the Lagrangian approach to find the angular frequency of small oscillations about the top. We can write the Lagrangian as:

L = T - U = (1/2)(m*v^2 + I*θ'^2) - mg[(r + b)cosθ + rθsinθ]

where v is the velocity of the center of mass, θ' is the angular velocity, and I is the moment of inertia of the cube about its center of mass. We can use the equations for the center of mass to express v and θ' in terms of θ, and then differentiate to find θ''. Plugging this into the Lagrangian, we can then use the Euler-Lagrange equation to find the equation of motion for θ. Solving this equation will give us the angular frequency of small oscillations about the top.

I hope this helps. Let me know if you have any further questions.