A materials testing laboratory simultaneously transmits four ultrasonic waves—labeled…
MATHEMATICS · Multi-part · A Level · Mathematics · Quantities and units in mechanics · Quantities and units in mechanics
Problem
A materials testing laboratory simultaneously transmits four ultrasonic waves—labeled D, E, F, and G—through the same metal rod to probe for internal flaws. A technician compiles the following table of measured quantities at a fixed point inside the rod:
Wave
Frequency (kHz)
Displacement Amplitude (μm)
D
150
0.80
E
75
1.60
F
300
0.40
G
225
0.80
Assume the average rate at which each ultrasonic wave transports energy is directly proportional to its frequency, while amplitude has no effect on that rate in this experiment.
(a) Rank the four waves from greatest to least displacement amplitude. Provide the ordered list and explain the basis of your ranking.
(b) Using the proportionality stated above, identify any pair of waves that transports the same average power through the rod. Show your reasoning.
(c) The technician wants Wave D to transport the same average power as Wave F by adjusting only its displacement amplitude. Analyze whether this goal is achievable under the given assumptions, and fully justify your conclusion.
(d) Suppose the assumption that amplitude does not affect power is later found to be invalid for very high amplitudes. Propose one experimental modification that would allow the technician to investigate how power actually depends on amplitude while holding frequency constant. Explain how your change isolates the desired relationship.