Who Will Win In a Downhill Race?

[physicStudent]

Homework Statement

Out of these various shapes: a wooden disk, a modified wooden disk with four pieces of metal in the middle, a smaller metal disk, a metal hoop, a small metal sphere, and a cart with four small wheels, who will win in a race rolling down a ramp with height h and angle Θ? An image of the cart is attached.

Homework Equations

v = \sqrt{\frac{2gh}{1+c}}, where c = coefficient of inertia

The Attempt at a Solution

We did an experiment in class where we were given the various shapes listed above (excluding the cart), and using the equation above, I found that the modified disk would win the race by thinking that the coefficient of its inertia must be smaller than that of the sphere (which came in second place with a coefficient of inertia of 2/5). However, I am stuck on this problem now with the inclusion of the cart. Any help is appreciated.

 

[Simon Bridge]

When an object goes downhill it exchanges gravitatonal potential energy for other forms - including translational kinetic energy.

If the object rotates, then some of the energy gets stored in the rotation.

For most of your objects, the whole thing is rotating ... so the equation you have should work well.
For the cart, only the wheels are turning ... so you need to derive a different equation.
It has been included to see if you understand what you are doing instead of just plugging numbers into a potted solution.

 

[physicStudent]

So what would change in the conservation of energy equation to help in deriving this new equation? I assume that I would still be looking for the final velocity of the cart as it reaches the bottom of the ramp, but I'm not sure as to how to include the non-rotating portion of the cart into the equation. Also, would the number of wheels change the equation in any way?

 

[physicStudent]

Looking at this site:

http://physics.stackexchange.com/qu...l-will-its-speed-depend-on-the-size-of-the-wh

They seemed to come up with an equation for the acceleration of what could be used for the cart as:

a = \frac{(M+Nm)gsin\theta}{M+Nm(1+c)}

Using this equation, I was wondering if I could compare the cart's acceleration with that of the modified disk to see who would win the race?

 

[Simon Bridge]

You know that gravitational potential energy is traded for linear kinetic energy and rotational energy - start by writing out an expression that says that for the cart.
Which parts of the cart rotate? Which translate?