Use the following figure to answer the following question:
-A steel wire and an aluminum wire of the same length L and diameter d are joined to form a wire of length 2L. The wire is fastened to the ceiling and a mass m is attached to the other end. If the masses of the wires are negligible relative to mass m, the ratio of the strain in the aluminum wire to the strain s in the steel wire is
A) less than 1.
B) equal to 1.
C) greater than 1.
D) dependent on mass m.
E) impossible to determine.

Question

Use the following figure to answer the following question:  
-A steel wire and an aluminum wire of the same length L and diameter d are joined to form a wire of length 2L. The wire is fastened to the ceiling and a mass m is attached to the other end. If the masses of the wires are negligible relative to mass m, the ratio of the strain in the aluminum wire to the strain s in the steel wire is
A) less than 1. 
B) equal to 1. 
C) greater than 1. 
D) dependent on mass m. 
E) impossible to determine.

Quizplus · Accepted Answer

Since the wires are of the same length and diameter but different materials, their Young's moduli are different. Let Ys be the Young's modulus of steel and Ya the Young's modulus of aluminum.

The strain in a material is given by:

strain = (change in length) / (original length)

Since the wire is fixed at one end and a mass is attached to the other end, the change in length of the wire will be proportional to the weight of the mass. Let F be the weight of the mass.

The change in length for the steel wire is given by:

delta_Ls = (F/2) / ((pi/4) * d^2 * Ys)

The change in length for the aluminum wire is given by:

delta_La = (F/2) / ((pi/4) * d^2 * Ya)

The ratio of the strains is:

strain_aluminum / strain_steel = delta_La / L / (delta_Ls / L)

After canceling out the length L and diameter d, we get:

strain_aluminum / strain_steel = Ya / Ys

Since Ya > Ys (Young's modulus of aluminum is greater than Young's modulus of steel), we have:

strain_aluminum / strain_steel > 1

Use the Following Figure to Answer the Following Question