How Is Energy Lost to Friction Calculated in a Pivoting Spool System?

[sullyjared]

Homework Statement

A spool of thin wire (with inner radius R=0.50m outer radius R=o.65m, and a moment of inertia I_cm=0.8957 kg*m^2) pivots on a shaft. The wire is pulled down by a mass M=1.5 kg. After falling a distance D=0.51m, starting from rest, the mass has a speed of v=64.5 cm/s. Calculate the energy lost to friction during that time.

Homework Equations

KE=.5Iw^2
KE=.5mv^2
v=rw
I=.5M(r^2+R^2)

The Attempt at a Solution

I have made two attempted solutions for this problem. The first was simple. I neglected the spool all together. Then I solved for the potential energy of the mass pulling down and the kinetic energy of this mass, and found the difference... that didn't work.
My second attempt was to calculate the kinetic energy of the spool with KE=.5Iw^2 and compare it to the Kinetic energy of the mass pulling down but to solve for the spools kinetic energy i need the moment of inertia about the axis of rotation, which means i need the mass of the spool, which i don't have... any suggestions?

 

[Mentz114]

I think you have to assume that the wire that unwinds does not change the MoI of the spool and wire. So your second method should work if you calculate w (omega) correctly. You have the outer dimension of the spool so that is straightforward.

 

[Moetasim]

I would suggest approaching this problem by considering the conservation of energy principle. The initial energy of the system (potential energy of the mass and rotational kinetic energy of the spool) should equal the final energy (kinetic energy of the mass and rotational kinetic energy of the spool). By solving for the initial and final energies, you can find the energy lost to friction during the time the mass falls a distance D.

To do this, you will need to use the equations you listed in your attempt at a solution, along with the fact that the initial angular velocity of the spool is 0 (since it starts from rest) and the final angular velocity can be calculated using the relationship v=rw. You can also use the moment of inertia equation that you listed to find the moment of inertia of the spool about the axis of rotation.

Once you have calculated the initial and final energies, the difference between them will give you the energy lost to friction. Keep in mind that friction will also cause the spool to rotate slower than it would without friction, so you will need to account for this in your calculations.

I hope this helps guide you towards a solution. Good luck!