Passage VI

A spring with a spring constant k (which indicates the spring’s stiffness) is attached to a cube of mass M and then to a wall. The cube initially rests at Point Q on a frictionless horizontal surface. The cube is pulled from Point Q to Point P, stretching the spring. Once released, the cube oscillates between Points P and R, each a distance A (the amplitude) from Point Q (see the diagram).

The frequency, f, is the number of times per second the cube completes a cycle (travels from P to R, then back to P); v is the average speed of the cube during 1 cycle.

The teacher asks each of 3 students to predict which combination of the independent variables k, M, and A would produce the highest f.

Student 1

As k increases, v decreases, because a stiffer spring moves less freely. Therefore, f decreases because it takes more time for the cube to complete each cycle.

As M increases, v decreases, and so f also decreases.

As A increases, v increases, because the distance the cube travels during a complete cycle increases. Therefore, f also increases.

The highest f would be produced when each of k and M has its least value and A has its greatest value.

Student 2

As k increases, v increases, because the force exerted on the cube by the spring also increases, giving the cube a greater average acceleration and, therefore, a greater v. When v increases, f increases.

As M increases, v and f increase, because during the spring’s oscillations, the momentum of the cube keeps its average speed from decreasing.

As A increases, the force exerted on the cube by the spring increases, so v, f, and average acceleration increase.

The highest f would be produced when each of k, M, and A has its greatest value.

Student 3

As k increases, the potential energy stored in the spring just before its release increases, so both v and f also increase.

As M increases, the cube’s average acceleration decreases, and so both v and f decrease.

As A increases, both the cube’s average acceleration and v increase, but so does the total distance through which the cube must travel. Consequently, f remains the same.

The highest f would be produced when k has its greatest value and M has its least value.