Rotational Motion: Who Reaches Bottom First?

[justagirl]

Two hoops of identical radii roll without slipping down identical inclined planes under the influence of gravity. Hoop 1 is identical to Hoop 2 except for some massive spokes which are fixed to the rim.

If the two hoops are released at the same time from the same height, which hoop makes it to the bottom of the ramp first?

Thanks!

 

[Justin Lazear]

Each hoop starts with the same amount of energy at the top of the ramp.

Which one has more translational kinetic energy at the bottom? Or, perhaps more importantly, which one has more rotational kinetic energy at the bottom, and how would this affect the translational KE?

What does this result mean for the time it takes to get to the bottom?

--J

 

[justagirl]

re;

The part of I'm confused on is this:

The wheel with the spokes has more mass, which increases inertia and decreases speed. But it also has more mass concentrated around the radius, which increases speed...

 

[Justin Lazear]

But it also has more mass concentrated around the radius, which increases speed...

What does this mean?

Consider the moment of inertia as a whole. Also remember that angular velocity (and hence angular KE) is proportional to linear velocity, i.e. the faster the wheel is moving down the ramp, the faster it's rolling.

--J

 

[justagirl]

I'm a lil confused:

so by conservation of energy I have:

mgh = 1/2mv^2 + 1/2Iw^2
mgh = 1/2mv^2 + 1/2I(v^2/r^2)

V^2(1/2m + 1/2(I/R^2) = mgh

v^2 = mgh / (1/2m + 1/2(I/R^2))

For spoke one, the mass is greater, so speed decreases...
But wouldn't I be smaller? Wouldn't that increase the speed?

 

[BobG]

The part of I'm confused on is this:

The wheel with the spokes has more mass, which increases inertia and decreases speed. But it also has more mass concentrated around the radius, which increases speed...

You've caught the important part of the problem.

If the total mass of the two hoops are equal, the wheel with the spokes has more mass towards the center and a smaller moment of inertia. It accelerates faster. (I=mr^2)

If the two hoops are identical (i.e. - they have the same amount of mass out on the rim) except one hoop has some additional mass added to it in the form of spokes, the one with the spokes has the larger moment of inertia. Technically, the moment of inertia is the sum of the moment of inertia of each piece of the hoop. So, adding mass, regardless of its placement, is adding more inertia.

In other words, read the question carefully and make sure which condition the question is talking about.

 

[justagirl]

would the smaller inertia hoop be going faster or does it not matter? I think we did this experiment where we found acceleration to be g sin theta / (1 + c), where c was the constant of inertia. (i.e., 1/2 for a cylinder and 1 for a hoop).

 

[justagirl]

help! can someone please answer/

 

[Justin Lazear]

The hoop with the spokes is both heavier and has a larger moment of inertia.

Suppose both hoops are going down the ramp with velocity v. Compare their total kinetic energies.

E = mv^2/2 + Iw^2/2

Note that both m and I are larger for the hoop with the spokes, so v and w must be smaller in order to achieve the same total energy.

--J