This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Identify the type of tube and the wave pattern. The diagram shows a tube that is closed at one end (bottom, indicated by shading) and open at the other end (top). For a stationary wave in a closed tube, there must be a node (point of zero displacement) at the closed end and an antinode (point of maximum displacement) at the open end. Step 2: Determine the relationship between the length of the air column (L) and the wavelength () from the diagram. The diagram shows the following pattern from the closed end to the open end: A node at the closed end. An antinode. Another node. An antinode at the open end. The distance from a node to an adjacent antinode is ()/(4). The distance from an antinode to an adjacent node is also ()/(4). Tracing the wave pattern in the diagram: From the node at the closed end to the first antinode: ()/(4) From the first antinode to the next node: ()/(4) From the second node to the antinode at the open end: ()/(4) Therefore, the total length L of the resonating air column is the sum of these segments: L = ()/(4) + ()/(4) + ()/(4) = (3)/(4) This corresponds to the first overtone (or third harmonic) for a closed tube. Step 3: Substitute the given wavelength and calculate L. The wavelength is given as = 40 \, cm. L = (3 × 40 \, cm)/(4) L = 3 × 10 \, cm L = 30 \, cm Step 4: Compare the result with the given options. The calculated length L = 30 \, cm matches option (d). The final answer is D. 30cm.