If the cars are racing side by side on a circular track, then they have the same angular velocity, because they complete their circles in the same amount of time. However, the car on the outside has a larger circle to complete, and thus must have a higher linear velocity.

For this problem the ball picks up rotational velocity as well as linear velocity. Keeping this in mind we first set up our equation of conservation of energy

Because the object starts at rest the initial kinetic energy is 0 and likewise the final potential energy is 0 because it has spent all its potential energy to become kinetic.

Where m is mass of the ball, g is gravity, h is height of the ramp, v is final velocity of the ball, I is the moment of inertia of the ball, and w is the final rotational velocity of the ball.

Now because the ball is a solid sphere we can represent I and w as follows

and

Plugging this into our equation of energy conservation

Now solve for velocity:

Now plugging in our value for height, we arrive at a final answer of 

This question deals with inertia. You can solve this mathematically, but the answer can also be found conceptually.

Inertia (in physics) is the object's ability to resist change in motion. In this case, it is based on mass, and how the mass of the object is distributed around it's rotational axis (Center of mass = middle of sphere). The larger the inertia, the more resistant to rotation; the more resistant to rotation, the slower the object will move down the incline.

Have you ever seen figure skating? When skaters want to make very tight turns or rotations, they pull their arms and head really close into their chest and crouch down into a smaller formation.

Why do skaters do this? Physics of course! The closer the mass of an object is to its COM, the smaller inertia they have, and the smaller resistance to change in motion. If the same skater (same height, weight, dimensions) attempts a turn in a further stretched out position (limbs untucked, etc), they will have a larger inertia, and therefore would not be able to turn as quickly as they could when tucked.

Okay so back to the spheres. As with the skaters, the further a mass is distributed from its COM, the larger the inertia. So between spheres A and B, which do you think has a lower inertia?

Sphere A has it's mass more evenly distributed throughout itself, whereas sphere B has the same mass, but with it's weight distributed solely to the exterior of its shell. Therefore, sphere A is a better choice that sphere B.

Now, what about sphere C? Typically, the larger the mass, the larger the inertia, same scenario for the objects' radii. The larger the M and R, the larger the inertia and the more resistant the object is to rotation. The best answer for Michelle to win is object A only. If you want to plug in some numbers to the inertia equations, I'll provide them here:

Displacement is the change in distance from start to finish

Where  is her position on the circular track. Since Laura starts and ends at the same place, the displacement is 0. It doesn't matter what we label her start/end position as, it's her initial position compared to her final.

Inertia is the ability for an object to resist motion. In this case, inertia is determined by how far away the mass (weights) are distributed from the dumbbell's COM (center of handle). While there is no definitive equation for the "inertia of a dumbbell", we can conceptualize this. All inertia equations deal with a mass and a radius. The larger either of these numbers are, the larger the inertia! We know that the larger the inertia, the more resistant that object is to change in motion (or the more awkward it would be to pick up that object). With that said, it would be harder for Steve to pick up the new weight. Not because of the weight itself, but because of how that same weight is distributed around that object. Think about it this way; a piece of paper weighs roughly .16 oz. if I gave you a 1 cm cube that weighed .16 oz, which would be easier to pick up and balance on your head? I would most likely assume that you would say the cube. This is because the weight of the cube is more closely positioned near the COM, rather than the thin sheet of paper with it's weight evenly spread throughout 8.5" x 11".

The bucket would proceed in the same direction as the tangential velocity at that given time. As seen in the figure below, each position of the bucket around the circle will give rise to a new tangential velocity vector. If the bucket is going counter clockwise and is released at the "3 o'clock" position, the  will be in the upward direction.