If a metal wire has its length doubled and its diameter tripled, by what factor does its Young ‘modulus change?
Young's is a property of the material of construction of a wire. It depends only on the nature of the material used for making the wire. Young's modulus does not depend on the physical dimensions such as the length and diameter of wire. Thus there will be no change in Young's modulus by merely doubling the diameter and tripling the length of a wire, without changing the basic nature of the material used for making the wire.
Please note that the Young's modulus depends on the chemical composition of the various ingredients present in the material as well as other treatment to which the material may have been exposed. To some extent the Young's modulus is also affected by temperature. Thus at very low temperature all materials tend to become rigid and their Young;s modulus tends to increase.