Simplifying Lagrange's Equations for a System of Connected Masses

[Quinner]

Two masses m and M are connected by a light inextensible string. (One mass is on a ramp of angle theta, and is connected to the other mass by a string going over a pulley and the other mass is hanging straight down) If the surface is frictionless, set up the equations of motion and find the acceleration of the system.

My question is this, am I making this too simple? Here's my kinetic energy function. 1/2m(dy/dt)^2 + 1/2M(dy/dt^2) and my potential is mgysin(theta) -Mgy. I'm pretty scatterbrained but it seems to me that there should only be one generalized co-ordinate so that is why I have set my equations up as such. But there's this little nagging part of my brain that says the kinetic energy has two components x and y, but I know energy is a scalar so maybe that's ludicrous. Any help would be appreciated.

 

[Quinner]

Homework Statement

Two masses m and M are connected by a light inextensible string. (One mass is on a ramp of angle theta, and is connected to the other mass by a string going over a pulley and the other mass is hanging straight down) If the surface is frictionless, set up the equations of motion and find the acceleration of the system.

Homework Equations

The Attempt at a Solution

My question is this, am I making this too simple? Here's my kinetic energy function. 1/2m(dy/dt)^2 + 1/2M(dy/dt^2) and my potential is mgysin(theta) -Mgy. I'm pretty scatterbrained but it seems to me that there should only be one generalized co-ordinate so that is why I have set my equations up as such. But there's this little nagging part of my brain that says the kinetic energy has two components x and y, but I know energy is a scalar so maybe that's ludicrous. Any help would be appreciated.

 

[Shooting Star]

The mass on the ramp has a x component of velo and the KE should have the term
m/2(dx/dt)^2 added to it. But you can eliminate x using the angle of the ramp.

 

[nrqed]

Two masses m and M are connected by a light inextensible string. (One mass is on a ramp of angle theta, and is connected to the other mass by a string going over a pulley and the other mass is hanging straight down) If the surface is frictionless, set up the equations of motion and find the acceleration of the system.

My question is this, am I making this too simple? Here's my kinetic energy function. 1/2m(dy/dt)^2 + 1/2M(dy/dt^2) and my potential is mgysin(theta) -Mgy. I'm pretty scatterbrained but it seems to me that there should only be one generalized co-ordinate so that is why I have set my equations up as such. But there's this little nagging part of my brain that says the kinetic energy has two components x and y, but I know energy is a scalar so maybe that's ludicrous. Any help would be appreciated.

What you wrote seems completely fine to me. There is only one generalized coordinate, you are correct.

 

[Shooting Star]

What you wrote seems completely fine to me. There is only one generalized coordinate, you are correct.

Err…I’m a bit puzzled here. What the OP has written is clearly incorrect. Are you referring to his post?

 

[nrqed]

Err…I’m a bit puzzled here. What the OP has written is clearly incorrect. Are you referring to his post?

Yes, I am referring to his post.
He is using a generalized coordinate y for his system (which has only one degree of freedom. The kinetic energy is simply 1/2 (m+M) (y dot)^2. This is a generalized coordinate, not a cartesian coordinate.

 

[Shooting Star]

(Forgot to reply.)

Yes, yes, nrqed, you are quite correct. I had misunderstood the OP’s problem at the first glance. I thought he was using Cartesian co-ordinates. Sorry Quinner. I hope you have solved it. Was your doubt clarified?