Passage V

Spheres W, X, Y, and Z, initially at rest at the starting line on an inclined plane, will be released one at a time and allowed to roll, without sliding. all the way to the finish line at the bottom of the incline. The masses and radii of Spheres W and X are compared in the following table, as are the masses and radii of Spheres Y and Z.

A teacher asks 2 students to predict whether Sphere W or Sphere X, if either, will reach the finish line in less time, and whether Sphere Y or Sphere Z, if either, will reach the finish line in less time.

Student 1

Sphere W will reach the finish line in less time than Sphere X. Sphere W weighs more than Sphere X, so the force of gravity exerted on Sphere W is greater than that exerted on Sphere X. Because force = mass x acceleration, Sphere W will have the greater acceleration down the incline and, therefore, the greater average speed.

Sphere Y will reach the finish line in less time than Sphere Z. For 2 spheres having the same mass, as sphere radius increases, sphere density decreases, so the energy required for a sphere to complete a rotation decreases. The mass of Sphere Y is spread over a greater volume than is the mass of Sphere Z, so Sphere Y is less dense thanSphere Z and will have the greater acceleration and the greater average speed.

Student 2

Spheres W and X will reach the finish line in the same amount of time. The acceleration of a sphere due to gravity is independent of the sphere's mass, so the 2 spheres will have the same acceleration and, therefore, the same aver-age speed.

Sphere Z will reach the finish line in less time than Sphere Y. For spheres having the same mass, as sphere radius increases, the energy required for a sphere to complete a rotation increases. This is because as sphere radius increases, the distance traveled by the sphere to complete a rotation increases. Therefore, because Sphere Y has the greater radius, it will have the lesser acceleration and the lesser average speed.