If the rods with diameters and lengths listed below are made of the same material, which will undergo the largest percentage length change given the same applied force along its length?a. d, 3L b. 3d, L c. 2d, 2L d. 4d, L

For this exercise we are asked to change the length of the bar by the action of a force applied along its length, in this case we focus on the expression of longitudinal elasticity

F / A = Y ΔL/L

where F / A is the force per unit length, ΔL / L is the fraction of the change in length, and Y is Young's modulus.

In this case the bars are made of the same material by which Young's modulus is the same for all

ΔL / L = (F / A) / Y

the area of the bar is the area of a circle

A = π r² = π d² / 4

A = π / 4 d²

we substitute

ΔL / L = (F / Y) 4 /πd²

changing length

ΔL = (F / Y 4 /π) L / d²

The amount between paracentesis are all constant in this exercise, let's look for the longitudinal change

a) values given d and 3L

ΔL = cte 3L / d²

ΔL = cte L /d² 3

To find the percentage, we must divide the change in magnitude by its value and multiply by 100.

ΔL/L % = [(F /Y 4/π 1/d²) 3L ] / 3L 100

ΔL/L % = cte 100%

b) 3d and L value, we repeat the same process as in part a

ΔL = cte L / 9d²

ΔL = cte L / d² 1/9

ΔL / L% = cte 100/9

ΔL / L% = cte 11%

c) 2d and 2L value

ΔL = (cte L / d ½ )/ 2L

ΔL/L% = cte 100/4

ΔL/L% = cte 25%

d) value 4d and L

ΔL = cte L / d² 1/16

ΔL/L % = cte 100/16

ΔL/L % = cte 6.25%

The highest percentage of change corresponds to the thinnest rod, the correct answer is a