Solve Rotational Problem: Find Disc Velocity as Function of Time

[MikkelR]

I'm kinda stuck in this mechanical problem, ill try to describe the situation.

A plane, homogeneous disk may slide on a frictionless horizontal table. A cord is fastened to the disc and wrapped many times around the rim of the disc. The cord goes without friction through a small eyelet at the edge of the table and is connected to a mass m. The mass of the disc is M and its radius is R. The system is started from rest and the cord is assumed to be taut throughout the motion. Ignore the mass of the cord.

http://ylle.eu/DSC00602.JPG (my sketch of the problem)

Describe the velocity of the disc as a function of time.

Here is what i have computed of equations so far.

The forces affecting the disc M must be M(d²x)/(dt²) = mg
The forces affecting the mass m must be m(d²x)/(dt²) = mg - S

The torque(N) exerted on M can be described as

N = mgR

The moment of inertia for a circular disk around its CM is

I = (1/2)MR²

The angular momentum is

L = Iw

Torque equals the change of angular momentum with respect to time

N = Iw'
N = (1/2)MR²w'
mgR = (1/2)MR²w'

And from this point i can't really seem to maky any more observations that can help me solve the problem.
I hope someone can clarify the problem for me

Thanks in advance
Mikkel

 

[OlderDan]

You need to rethink the force acting on the disk. Look at what you did for the hanging mass and think about the connection between the two objects.

 

[m2003]

I would suggest that you continue to use the equations and observations you have made so far to solve the problem. You have correctly identified the forces and torques acting on the disc and mass, and have calculated the moment of inertia and angular momentum. These are all important pieces of information that will help you find the disc's velocity as a function of time.

One approach you could take is to use the equations of motion for rotational motion, which state that the torque equals the moment of inertia times the angular acceleration. You have already found an expression for the torque, so you can use that to solve for the angular acceleration, which can then be integrated to find the angular velocity as a function of time.

Another approach could be to use conservation of energy. At the start of the motion, the disc and mass both have zero kinetic energy and are at rest. As the system moves, energy will be transferred between the disc and the mass. By setting up an energy conservation equation, you can solve for the velocity of the disc at any given time.

Overall, it may be helpful to break down the problem into smaller steps and use the equations and concepts you have already identified to solve for the disc's velocity as a function of time. Good luck!