Final answer:
The full circumference of a pole is typically tested when conducting a hammer test to evaluate its integrity. However, the provided scenario is about resonance in a lab pole, which falls under physics. The lowest resonant frequency depends on the physical properties of the pole and the point of excitation.
Step-by-step explanation:
The question regarding the range circumferentially that should be tested when hammer testing a pole seems to be part of a general inquiry rather than related to the specific scenario provided, which is about resonance frequency in a lab pole.
Nonetheless, to determine the integrity and condition of the whole pole, full circumference testing is typically done. However, the given scenario about the resonant frequency of an aluminum lab pole involves an applied concept in physics, specifically wave mechanics in solids.
A lab pole, being a solid object, can resonate when struck. If the student is holding the pole one quarter of the length from the end and strikes it, we are essentially forming a node at that point and creating an antinode at the free end, which leads to a standing wave.
The lowest frequency at which this would occur is the fundamental frequency or the first harmonic of the system. Without additional information such as the speed of sound in aluminum or the tension in the pole, we cannot provide a specific numerical frequency.
Yet, the first harmonic frequency can be calculated using the formula f = v / (2L), where v is the speed of sound in aluminum and L is the length of the segment of the pole that vibrates - in this case, three-fourths of the pole since the pole is fixed at the point where the student is holding it. This is a basic concept in the physics of waves and resonance.