How to Solve These Rotational Dynamics Problems?

[csnsc14320]

1. A wheel of mass M and radius of gyration "k" spins on a fixed horizontal axle passing through its hub. Assume that the hub rubs the axle of radius "a" at only the top-most point, the coefficient of kinetic friction being μk. The wheel is given an initial angular velocity Wo. Assume uniform deceleration and find (a) the elapsed time and (b) the number of revolutions before the wheel comes to a stop.

2. A solid cylinder of length L and radius R has a weight W. Two cords are wrapped around the cylinder, one near each end, and the cord ends are attached to hooks on the ceiling. The cylinder is held horizontally with the two cords exactly vertical and is then released. Find (a) the tension in each cord as they unwind and (b) the linear acceleration of the cylinder as it falls.





For the first question, I'm not really sure where to start, and we didn't cover radius of gyration. I think if I just get a set up of what information to work with I can get this.

The second one, I drew a FBD and I see that the two tensions will be exerting a torque on the cylinder, but I don't think that the weight gives any torque given where it's acting from - but then I don't know how to incorporate it into getting my answer. I have that 2TR + ?? =I*a/r so far =\

Thanks for any help to get nudge me in the right direction :D

 

[chrisk]

The radius of gyration of an object is the radius if all the mass were concentrated at that radius to give the same moment of inertia of the object

I = Mk²

To start Problem 1 use the work-energy theorem, the rotational kinematic relation

Work = Torque*theta

and the fact that the wheel will initially and very quickly move off the high point of the axle due to friction and reach an equilibrium position. Utilize the angle of repose to determine the equilibrium position.

For Problem 2, the weight, W, does impart a torque since T = r x Mg = r x W . If the strings were attached over the center of the cylinder then no torque would exist.

 

[csnsc14320]

The radius of gyration of an object is the radius if all the mass were concentrated at that radius to give the same moment of inertia of the object

I = Mk²

To start Problem 1 use the work-energy theorem, the rotational kinematic relation

Work = Torque*theta

and the fact that the wheel will initially and very quickly move off the high point of the axle due to friction and reach an equilibrium position. Utilize the angle of repose to determine the equilibrium position.

For Problem 2, the weight, W, does impart a torque since T = r x Mg = r x W . If the strings were attached over the center of the cylinder then no torque would exist.

For problem one, we haven't yet covered the work-energy theorem for rotational motion (unless we are suppose to infer that for ourselves), is there any other way to go about solving it? Or should I just look ahead and try it that way?
Also, I'm not sure if the axle fits perfectly through the wheel's hub, but if it did, why would the wheel move from the high point due to friction? What I imagine seeing is the hub being slightly larger and the inner radius of it to move around the circumference of the axle? (but then the wheel would be wobbling =\)

For problem 2, I used Iα=2TR, and using α=a/R, I=(MR^2)/2 got that T=Ma/4
For the acceleration I used Newtons 2nd law, and got Mg - 2T = Ma --> Mg -Ma/2 = MA --> a = 2g/3 (which makes T = Mg/6

 

[chrisk]

The work-energy theorem for linear motion should have been previously covered. Just use the analagous rotational kinematic equations to establish the equations. The hub is a little larger than the axle diameter so a slight shift off of vertical will occur.

Use the work-energy theorem for Problem 2. Solving Problem 1 first will give you the insight to solve Problem 2. Just keep in mind that two types of energy exist for the second problem; linear and rotational.