The Young’s modulus of a steel wire of radius r and length l is Y. If both the radius and length of the wire are doubled, what happens to the Young’s modulus?
Young’s modulus is a material property that measures the stiffness or elasticity of a material. It is defined as the ratio of stress to strain.
Y = Stress / Strain
Stress is the force applied per unit area and strain is the change in length divided by original length. Thus:
Y = (F/A) ÷ (ΔL/L)
Young’s modulus depends only on the nature of the material and not on the dimensions of the object. Whether the wire is thin or thick, short or long, the value of Young’s modulus remains the same as long as the material is unchanged.
In this question, the wire is made of steel before and after changing its dimensions. The radius is doubled and the length is also doubled. These changes affect the cross-sectional area and total elongation under load, but they do not change the intrinsic elastic property of steel.
Since Young’s modulus depends solely on the material composition and atomic structure, altering the dimensions cannot affect its value.
Therefore:
Hence, the Young’s modulus of the wire after doubling both radius and length remains exactly the same.
Young’s Modulus = Y (Unchanged)