Five objects of the same mass M: a hoop (radius R), a solid sphere (radius R), a

Question

Five objects of the same mass M: a hoop (radius R), a solid sphere (radius R), a thin-hollow sphere with the same radius as that of the sphere (radius R), a solid cylinder (radius R), and a small point-like particle, all race down a ramp of height h and slope (45 degree). The frictions are just large enough to cause the first four objects to roll, but is negligible to the particle so it comes down the ramp friction free (approximately). Determine the rank of the race: the hoop_______, the solid sphere_______, hollow sphere_______, solid cylinder_______and the particle_______.

(A) First, (B) second, (C) third, (D) fourth, (E) fifth, (F) Cannot be determined, (G) None of these.

Explanation / Answer

If you "race" these objects down the incline, they would definitely not tie! This is because Newton's Second Law for Rotation says that the rotational acceleration of an object equals the net torque on the object divided by its rotational inertia. (Net torque replaces net force, and rotational inertia replaces mass in "regular" Newton's Second Law.) The net torque on every object would be the same - due to the weight of the object acting through its center of gravity, but the rotational inertias are different. This means that the solid sphere would beat the solid cylinder (since it has a smaller rotational inertia), the solid cylinder would beat the "sloshy" cylinder, etc. The hoop would come in last in every race, since it has the greatest moment of inertia (resistance to rotational acceleration).

It turns out, that if you calculate the rotational acceleration of a hoop, for instance, which equals (net torque)/(rotational inertia), both the torque and the rotational inertia depend on the mass and radius of the hoop. This means that both the mass and radius cancel in Newton's Second Law - just like what happened in the falling and sliding situations above! (The mathematical details are a little complex, but are shown in the table below) This means that all hoops, regardless of size or mass, roll at the same rate down the incline! The same is true for empty cans - all empty cans roll at the same rate, regardless of size or mass. However, every empty can will beat any hoop! All solid spheres roll with the same acceleration, but every solid sphere, regardless of size or mass, will beat any solid cylinder! Cool, huh?

A circular object of mass m is rolling down a ramp that makes an angle theta with the horizontal. The weight, mg, of the object exerts a torque through the object's center of mass. The object rotates about its point of contact with the ramp, so the length of the lever arm equals the radius of the object.

rolling on a ramp diagram

Now, the component of the object's weight perpendicular to the radius is shown in the diagram at right. This means that the torque on the object about the contact point is given by:

and the rotational acceleration of the object is:

where I is the moment of inertia of the object. Now the moment of inertia of the object = kmr2, where k is a constant that depends on how the mass is distributed in the object - k is different for cylinders and spheres, but is the same for all cylinders, and the same for all spheres. The rotational acceleration, then is:

rotational acceleration equation

So, the rotational acceleration of the object does not depend on its mass, but it does depend on its radius. However, we are really interested in the linear acceleration of the object down the ramp, and:

linear acceleration equation

This result says that the linear acceleration of the object down the ramp does not depend on the object's radius or mass, but it does depend on how the mass is distributed. Therefore, all spheres have the same acceleration on the ramp, and all cylinders have the same acceleration on the ramp, but a sphere and a cylinder will have different accelerations, since their mass is distributed differently. All spheres "beat" all cylinders. All cylinders beat all hoops, etc.

so the order is fourth,first,second,third,fifth

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