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Young's modulus is an essential mechanical property that describes a material's stiffness. It is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. In essence, Young's modulus tells us how difficult it is to stretch or compress a material.
Young's modulus is represented by the symbol \( E \), and it is expressed in units of pressure, namely pascal (N/m²). A higher Young's modulus indicates a stiffer material, which deforms less under the same applied force compared to a material with a lower Young's modulus.
Knowing the Young's modulus of a material is crucial when you're dealing with mechanical engineering and civil project applications, such as construction, where understanding material properties can impact safety and function.
In the case of the exercise, the metal rod has a Young's modulus of \( 1.20 \times 10^{10} \) N/m². This value means the metal is quite rigid, which implies that it does not easily deform under stress.

Density is a fundamental property that signifies how much mass is contained in a unit volume of a substance. The formula to calculate density is \( \rho = \frac{mass}{volume} \), where \( \rho \) is the density. It is expressed in terms of kilograms per cubic meter (kg/m³).
Understanding the density of a material is important in contexts ranging from buoyancy analysis to determining material quality. For example, in construction, materials with high density might be preferred for certain load-bearing applications.
In the provided exercise, the material of the rod has a density of \( 8920 \) kg/m³. This high density suggests the material is substantial and relatively weighty, containing a lot of mass in a small volume—reflecting its robust nature.
When calculating wave speed in a solid, considering density alongside Young's modulus allows the examination of how mass distribution within the solid impacts wave propagation.

Compression waves, also known as longitudinal waves, are a type of mechanical wave where particle displacement is parallel to the direction of wave propagation. These waves can travel through solids, liquids, and gases. In solids, compression waves are often referred to as sound waves.
In the context of solids, the speed of these waves is significantly influenced by the material's properties, specifically Young's modulus and density. The wave speed \( v \) in a solid is given by the formula \( v = \sqrt{\frac{E}{\rho}} \), where \( E \) is the Young's modulus and \( \rho \) is the density.
This equation tells us that the wave speed is proportional to the square root of the stiffness of the material (Young's modulus) and inversely proportional to the square root of its density. Thus, a material with high stiffness and low density allows waves to travel faster.
In the exercise, using the values \( E = 1.20 \times 10^{10} \) N/m² and \( \rho = 8920 \) kg/m³, the calculated wave speed turns out to be approximately \( 36.68 \) m/s. This speed indicates how quickly compression waves traverse through the specified metal rod.