Physics
Physics
Problem
An aluminium wire of cross-sectional area is joined to a steel wire of the same cross-sectional area. This compound wire is stretched on a sonometer pulled by a weight of 10 kg. The total length of the compound wire between the bridges is 1.5 m of which the aluminium wire is 0.6 m and the rest is steel wire. Transverse vibrations are setup in the wire by using an external source of variable frequency. Find the lowest frequency of excitation for which the standing waves are formed such that the joint in the wire is a node. What is the total number of nodes at this frequency? The density of aluminium is and that of steel is ( ).Intuitive explanation
The key insight is that standing waves on a compound wire require the joint to be a node, which means each segment must independently support an integer number of half-wavelengths. Since the two materials have different wave speeds (due to different densities), they'll have different wavelengths at the same frequency—so you need to find the lowest frequency where both segments fit an integer number of half-waves with the joint as a boundary node. Calculate wave speed in each material using (where is linear mass density), then apply the node condition to both segments simultaneously.
Intuitive explanation
The key insight is that standing waves on a compound wire require the joint to be a node, which means each segment must independently support an integer number of half-wavelengths. Since the two materials have different wave speeds (due to different densities), they'll have different wavelengths at the same frequency—so you need to find the lowest frequency where both segments fit an integer number of half-waves with the joint as a boundary node. Calculate wave speed in each material using (where is linear mass density), then apply the node condition to both segments simultaneously.