Axial elongation: uniformly tapering circular bar vs uniform bar The extension of a circular bar tapering uniformly from diameter d1 at one end to diameter d2 at the other end, under axial load P, is ______ the extension of a uniform circular bar having diameter √(d1 * d2) subjected to the same load P and length.

Introduction / Context:
Bars with varying cross-section are common in weight-optimized designs. For an axially loaded, uniformly tapering circular bar, engineers often compare its elongation with that of a uniform bar to develop quick checks and design equivalences.

Given Data / Assumptions:

• Prismatic bar of length L with circular cross-section tapering linearly from d1 to d2.
• Axial load P within elastic limit; Young’s modulus E is constant.
• Comparison bar has constant diameter equal to √(d1 * d2) and the same length L and load P.

Concept / Approach:
Differential elongation is dδ = (P dx)/(E A(x)). For a taper, area A(x) varies with diameter d(x). Integrating along the length yields a closed-form expression. A well-known result is that a linearly-tapered circular bar elongates by the same amount as a uniform bar whose diameter equals the geometric mean of the end diameters.

Step-by-Step Solution:
For taper: d(x) varies linearly, so A(x) = π [d(x)]^2 / 4.δ_taper = ∫[0→L] P / (E A(x)) dx = (4 P / (π E)) ∫[0→L] 1 / [d(x)]^2 dx.Carrying out the integral for linear diameter variation gives δ_taper = (4 P L) / (π E d1 d2).For uniform bar with diameter d_u = √(d1 d2): δ_uniform = (4 P L) / (π E d_u^2) = (4 P L) / (π E d1 d2).Therefore δ_taper = δ_uniform, i.e., the extensions are equal.

Verification / Alternative check:
Dimensional consistency and limiting checks (d1 = d2 ⇒ both reduce to the same uniform-bar formula) confirm the equality.

Why Other Options Are Wrong:
Less than/greater than contradict the closed-form integral. “Indeterminate” is incorrect because sufficient information is given. “Zero” would require P = 0 or infinite stiffness, neither applicable.

Common Pitfalls:
Using arithmetic mean instead of geometric mean; integrating area rather than its reciprocal; forgetting that E cancels in the comparison.

Final Answer:
equal to