Scientists are measuring the properties of a newly discovered elastic material. They create a 1.5-m-long, 1.6-mm-diameter cord, attach an 850 g mass to the lower end, then pull the mass down 2.5 mm and release it. Their high-speed video camera records 36 oscillations in 2.0 s. What is Young's modulus of the material?

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Determine the frequency of oscillation. The frequency \( f \) is the number of oscillations per second. Use the formula \( f = \frac{N}{t} \), where \( N \) is the number of oscillations (36) and \( t \) is the time (2.0 s).

Calculate the angular frequency \( \omega \) using the relationship \( \omega = 2 \pi f \). This will help us relate the oscillatory motion to the material's properties.

Relate the angular frequency \( \omega \) to the spring constant \( k \) using the formula \( \omega = \sqrt{\frac{k}{m}} \), where \( m \) is the mass (850 g converted to kilograms). Solve for \( k \).

Use the relationship between the spring constant \( k \) and Young's modulus \( Y \). For a cylindrical cord, \( k = \frac{Y A}{L} \), where \( A \) is the cross-sectional area of the cord (calculated using \( A = \pi r^2 \), with \( r \) being the radius of the cord) and \( L \) is the length of the cord (1.5 m). Solve for \( Y \).

Substitute all known values into the formula for \( Y \) and simplify. Ensure all units are consistent (e.g., meters, kilograms, seconds) before performing the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Young's Modulus

Young's modulus is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain. It quantifies how much a material will deform under a given load, providing insight into its elastic properties. A higher Young's modulus indicates a stiffer material that deforms less under stress.

Oscillation Frequency

The oscillation frequency refers to the number of complete cycles of motion that occur in a unit of time. In this context, it can be calculated from the number of oscillations recorded and the time interval. Understanding frequency is crucial for analyzing the dynamic behavior of the elastic material when subjected to forces.

Tensile Stress and Strain

Tensile stress is the force applied per unit area of a material, while tensile strain is the deformation experienced by the material relative to its original length. These concepts are essential for calculating Young's modulus, as they provide the necessary parameters to understand how the material responds to applied forces.

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