Rotational Motion: Calculating Angular Speed of Disk and Neutron Star Spin Rate

[jakeowens]

A solid cylinder, a solid sphere, and a hoop all of the same mass but different radii, roll without sliding down an inclined plane. The body that gets to the bottom first will invariably be:
the cylinder
the sphere
they all arrive together
not enough information
the hoop

There isn't enough information to solve that problem is there? they all have different radii, so there is no way you can say which one will arrive there the first, because there just isn't enough information right?

Calculate the angular speed of a uniform solid disk about its central symmetry axis if its rotational kinetic energy is 86 J and its moment-of-inertia is 9.0 kgm2.

For this problem i just used the equation that angular KE=1/2Iw^2. I substituted my values for KE and I, and came up with w=4.37 rad/s. is that right?

t is possible for a large star (one greater than 1.4 solar masses) to gravitationally collapse, crushing itself into a tiny neutron star perhaps 40 km in diameter. Suppose such a thing happens to a star which is 2.09e9 m in diameter and spinning at a rate of about once around every 24 days. What would be the spin rate for the resulting neutron star?

For this last problem, i just assumed the ratio of the diameter to the days it takes to make a revolution would be the same. Is that right? or do i need to use some formulas on this one. It seemed incredibly easy the way i did it.


Thanks

 

[Diane_]

I'd agree with the first two. For the last one, you're dealing with the conservation of angular momentum. The moment of inertia of the star will change as it shrinks, with a corresponding increase in the angular speed. Moment of inertia is quadratic in the radius, so cutting the radius in half while keeping the mass the same would result in a four-fold change in the moment of inertia. Since angular momentum is linear in angular speed, you won't be able just to take a one-to-one ratio between the radius of the star and the angular speed. The problem isn't very much harder than what you did, but it is a little.

 

[Fermat]

For the first one, there actually is enough information.
The (linear) acceleration of a rolling object going down a plane depends on the angle of inclination and also the inertia of the object.

There are two parts to working out this acceleration.
Part 1.
What are the forces acting on the object as it rolls down the plane.
Use Newton's 2nd law to get an eqn of motion.
Part 2.
What forces on the object provide rotational motion on the object.
Use the rotational equivalent of Newton's 2nd law to get another eqn of motion.

Combine these two eqns and simplify to get the linear acceleration of each object. Rank the accelerations to find out which one arrives first.