Question

Critical Thinking Two spherical beads of the same mass fall at a constant speed due to a drag force. Bead A takes twice as long to fall the same distance as bead B. (a) Which of these beads has the larger radius? (b) Based on experimental data, an equation that fits the data is suggested for radius of a bead, r, which may not be correct: r=Kt−12, where K is a constant with appropriate units and t is the time it takes for a bead to reach a determined distance. Is this equation consistent with your answer from part a? Explain why or why not. Does this equation make sense? Explain why or why not. (c) Beads of different radii are selected and the time it takes to fall a set distance is recorded. Graph the radius of these beads as a function of the time it takes to fall a determined distance.Which of these beads has the larger radius?

a) Bead A
b) Bead B
c) Both have the same radius
d) It depends on the drag force

Answer 1

Bead A has a larger radius as it falls slower due to greater drag force. The provided equation r = Kt - 1 is not consistent as it implies an inverse relationship between radius and time, contradicting the scenario that a larger radius increases drag force and thus time to fall.

Step-by-step explanation:

Critical Thinking: Drag Force and Spherical Beads

Bead A has the larger radius compared to Bead B as they are of the same mass and fall at a constant speed due to drag. The drag force acting on the beads is dependent on their cross-sectional area, which in turn depends on the radius of the beads. A larger radius implies a larger cross-sectional area, which increases the drag force. Since Bead A takes twice as long as Bead B to fall the same distance, it is experiencing a larger drag force, indicating that it has a larger radius.

The suggested equation r = Kt - 1, where K is a constant and t is the time taken to fall a determined distance, implies that the radius decreases as time increases, which is not consistent with our understanding that a larger radius should cause the bead to fall more slowly due to an increased drag force. Therefore, this equation does not make sense as it suggests radius is inversely proportional to time, which contradicts the scenario provided.

To graph the radius as a function of time based on our understanding, we would expect to see an increasing function, where the radius increases as the time taken to fall increases. If we were to plot this, the y-axis would represent the radius r, and the x-axis the time t. According to our analysis, this graph should show a positive correlation between the radius of the beads and the duration of their fall. However, the proposed equation would yield a graph that shows a negative correlation, which is incorrect based on the given information.