Why does a more center focused mass have a larger angular velocity?

I understand that the angular velocity increases if the radius is decreased because w(omega)=v/r.





I also understand that if the radius is decreased, the moment of inertia decreases because I=mr^2.





I'm a bit confused with this question I had in a review. It asked which wheel would roll down an incline first. A heavy tire/light hub or light tire/heavy hub.The tire with the heavy hub rolls to the bottom first.





Could someone explain to me why this is, preferably with equations to back it up? Also, what does the translational and rotational motion have to do with it?|||The first thing to realize here is that the r in w=v/r and the r in I=mr^2 actually refer to different things. Angular velocity depends only on the outer radius of the rolling object, so the first r doesn't change between the two tires. This means that w is proportional to v. On the other hand, moment of inertia depends on the distribution of mass at various radii, so the second r is smaller in the heavy-hub wheel than in the heavy-tire wheel. Mass is constant in both cases, so I is proportional to r^2.





Next, recall that R = 1/2*I*w^2, where R is rotational kinetic energy. By the arguments above, we see that R is proportional to v^2*r^2. That means that at a given velocity, the heavy-tire wheel will have more rotational kinetic energy since it has a larger r. That extra energy has to come from somewhere; in this case, the wheel must roll further down the incline and turn more gravitational potential energy into kinetic energy to achieve a given velocity. Since the heavy-tire wheel must roll further than the heavy-hub wheel to gather up enough energy to achieve a given velocity, it will be rolling more slowly than the heavy-hub wheel at any given point on the incline, and it will take longer to reach the bottom.|||you already gave the answer.





inertia is a *resistance* to an applied force. more inertia -%26gt; more resistance.





i.e. a larger (moment of) inertia is more resistant to an acceleration by an applied force.





the heavy tire/light rim has a larger moment of inertia than a light tire/heavy rim, simply because more mass is concentrated at lower radii in the latter case. i.e. if the total mass of the tire+rim is the same in both cases, the moment of inertia of the latter case is smaller than the moment of inertia in the former case because a larger fraction of the total mass is concentrated at lower radii.





and as you noted, the moment of inertia involves a factor that goes like m*r^2...





cheers|||think of the figure skater spinning with her leg out. she is spinning slowly. Then she pulls her leg in and starts spinning faster. In the first case she is a heavy tire light hub, in the second, she is light tire /heavy hub.





if the figure skater doesn't work for you, try this: find an office chair that spins around. sit in it and spin it and keep your legs out. then pull your legs in and feel the increase in angular velocity.





The more mass concentrated in the center, the more omega increases to balance the equation.