How Many Degrees of Freedom Does a Ball Rolling Down a Wedge Have?

[pentazoid]

Homework Statement

A uniform ball of mass m rols down a rough wedge of mass M and angle alpha , which itself can slide on a smooth horizontal table. The whole system undergoes planar motion. How many degrees of freedoms does this system have.? Obtain langrange's equations. for special case in which M =3m/2, find (i) the acceleration of the wedge, ii) the acceleration of the ball releative to the wedge.

Homework Equations

The Attempt at a Solution

I am not going to derived my Lagrange equations. I just want know if I have calculated to correct kinetic energy for the wedge and ball and the correct potential energy for the wedge and ball.

T_total=T_ball+T_wedge=T_lin+T_rot.+T_wedge

T_wedge=.5*(3m/2)*v_x^2

v_ball=v_xi+(a*cos(alpha)i+a sin(alpha)k)dtheta/dt

T_ball=.5*m*v_ball^2+.5*(2/5*m*a^2)(*v_ball^2)/a^2. They don't actually tell you the radius of the ball, but I assumed they wanted us to assigned the radius of the ball to a letter and I chose my radius to be a. The first term represents the ball's linear motion and the second term represents the ball's rotational motion.

 

[Dr.D]

You have not shown any diagram, but just from looking at what you have written, I think you have the kinematic relations incorrect. Until the kinematics are correctly established, the energies cannot possibly be written correctly.

You need to go back to the first question that asked how many DOF does this system have? That will tell you something about how many variables you are going to need to set up to write your energies correctly.

This is a pretty good problem. You can learn a lot from it if you do it correctly, but you will have to think.

 

[pentazoid]

You have not shown any diagram, but just from looking at what you have written, I think you have the kinematic relations incorrect. Until the kinematics are correctly established, the energies cannot possibly be written correctly.

You need to go back to the first question that asked how many DOF does this system have? That will tell you something about how many variables you are going to need to set up to write your energies correctly.

This is a pretty good problem. You can learn a lot from it if you do it correctly, but you will have to think.

since the ball is not changing heights, I say only the x and y coordinates are changing. Therefore, it should have two degrees of freedom.. Should their be an equation for r_1 and r_2

 

[Dr.D]

The problem statement said, "ball of mass m rols down a rough wedge..." so how can you say the ball is not changing heights if it rolls DOWN?

You say, "only the x and y coordinates are changing." And that would be the x and y coordinatess of what?

Third, repeat this process until you are out of things to locate and coordinates to choose.

Now, write the position of the CM of the ramp in terms of those coordinates required to locate the ramp, write the position of the CM of the ball in terms of those coordinates required to locate the ball, etc., all referred to a fixed, global coordinate system.
You mention and r_1 and an f_2, but I have no idea what thiese might be.

You need to approach the problem this way.

First, you know that there is a ramp, the rough wedge, that can slide on the smooth table. How many coordinates will it take to tell you where the ramp is located, and what would be a convenient choice? A coordinate may be measured parallel to x or y, but it does not have to be. It can be anything (a distance, angle, etc.) that measures where something is located.

Second, you know that there is a ball that rolls on the ramp. How many coordinates will it take to locate the ball relative to the ramp, and what would be a good choice?

Third, repeat this process until you run out of masses to be located and coordinates to be choosen.

Then write the position vector of the CM for each body in terms of the coordinates that locate that CM, as referred to a fixed, global coordinate system. When you differentiate this position vector, you have the velocity vector for the body, etc. Then you should be off and running.