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Understanding the tension in a wire is critical when studying the fundamental frequency of vibration. Tension, noted as \( T \), is the force applied along the wire, pulling it tight. • The tension directly affects the wave speed on the wire, which in turn influences the frequency.• Higher tension generally leads to higher frequency because the wave traverses the wire faster.
In the original exercise, the tension is given in a ratio of \(2:1\). This means that the tension in the first wire is twice that of the second wire. This difference in tension is a decisive factor when determining the frequency ratio of the two wires.

Linear mass density is another vital concept when analyzing the frequency of a vibrating wire. This density is denoted by \( \mu \) and defined as the mass per unit length of the wire.• It is calculated using the formula \( \mu = \rho \pi r^2 \), where \( \rho \) is the density and \( r \) is the radius.• Linear mass density affects the frequency inversely; higher density tends to lower the frequency because it slows down the wave speed.
In our case, linear mass densities are derived from the given ratios of density and radii. For the first wire, this results in \( 9\rho \pi r^2 \) due to the larger radius, while for the second wire, it’s \( 2\rho \pi r^2 \) with a smaller radius and higher density.

The density ratio given in the problem helps us understand how material properties of the wire affect the frequency.• Density, \( \rho \), relates to how much mass is in a given volume. It impacts the linear mass density of the wire, which we have seen affects the wave speed and therefore the frequency.
The original conditions present the density ratio as \(1:2\). The first wire has a density \( \rho \) while the second wire has \( 2\rho \). The second wire's higher density contributes to a higher linear mass density, making it vibrate at a lower frequency compared to an identical wire with less density.

The frequency ratio is the ultimate goal of our calculations, providing insight into how different properties interact to affect wave motion.• The formula for frequency \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \) intertwines tension and linear mass density.• The higher the tension and the lower the mass density, the higher the frequency of vibration.In the problem, we derive the frequency ratio by using the given tension and linear mass density ratios. After the detailed math steps, the ratio of fundamental frequencies \( f_1 : f_2 \) results in \( 2 : 3 \). This signifies that the first wire, with higher tension and lower effective density, vibrates at a higher fundamental frequency compared to the second wire.