A very light 1.00-m wire consists of two segments of equal length, one of steel (Young's modulus is 2.00 × 10¹¹ N/m²) and one of brass (Young's modulus is 9.0 × 1010 N/m²). The steel segment is 1.50 mm in diameter, and the brass segment has twice this diameter. When a weight w is hung from the ceiling by this wire, the steel segment stretches by 1.10 mm. By what distance does the brass segment stretch? A) 0.50 mm B) 0.61 mm C) 1.2 mm D) 2.4 mm E) 9.8 mm

Question

Quizplus · Accepted Answer

First, we need to find the effective Young's modulus of the wire. Since the wire consists of two segments of equal length, the effective Young's modulus can be found using the formula:

1/E_total = 1/ E_steel + 1/ E_brass

Substituting the given values, we get:

1/E_total = 1/(2.00 × 10^11 N/m^2) + 1/(9.0 × 10^10 N/m^2)

Solving for E_total, we get:

E_total = 1.38 × 10^11 N/m^2

Next, we need to find the cross-sectional area of the brass segment. Since its diameter is twice that of the steel segment, its radius is also twice that of the steel segment. Therefore, its cross-sectional area is:

A_brass = π(2r)^2/4 = 4πr^2 = 4π(0.75 × 10^-3 m)^2 ≈ 1.77 × 10^-6 m^2

Now, we can use the formula for elongation:

ΔL = FL/ AΕ

where F is the weight hanging from the wire, L is the length of the wire, A is the cross-sectional area of the wire, and Ε is the Young's modulus of the wire.

Since the length and cross-sectional area are the same for both segments of the wire, we can write:

ΔL_steel = F/2AΕ_steel

ΔL_brass = F/2AΕ_brass

Substituting the given values, we get:

1.10 × 10^-3 m = F/(2π(0.75 × 10^-3 m)^2(2.00 × 10^11 N/m^2))

ΔL_brass = F/(2π(1.5 × 0.75 × 10^-3 m)^2(1.38 × 10^11 N/m^2))

Solving for F and substituting the second equation into the first one, we get:

F = (2.00 × 10^11 N/m^2)(1 π(0.75 × 10^-3 m)^2)(1.10 × 10^-3 m) / (2π(1.5 × 0.75 × 10^-3 m)^2 + 4π(0.75 × 10^-3 m)^2(1.38 × 10^11 N/m^2))

F ≈ 699 N

ΔL_brass = (699 N)/(2π(1.5 × 0.75 × 10^-3 m)^2(1.38 × 10^11 N/m^2))

ΔL_brass ≈ 0.61 mm

Therefore, the answer is B.

A Very Light 1