The relationship L = Li + LiT is a valid approximation when T is small. If T is

Question

The relationship L = Li + LiT is a valid approximation when T is small. If T is large, one must integrate the relationship dL = L dT to determine the final length.

(a) Assuming the coefficient of linear expansion of a material is constant as L varies, determine a general expression for the final length of a rod made of the material. (Use any variable or symbol stated above as necessary.)

(b) Given a rod of length 1.35 m and a temperature change of 120.0°C, determine the error caused by the approximation when = 2.00 10-5 (°C)-1 (a typical value for a metal).

(c) Determine the error caused by the approximation when = 0.020 0 (°C)-1 (an unrealistically large value for comparison)

Explanation / Answer

a) dL = L dT

dL/L = dT

integrating both sides we have

ln L from Li to Lf = T from Ti to Tf

ln (Lf / Li) = (Tf - Ti)

Lf = Li*e^((Tf - Ti)

b) Lf = 1.35*e^(2*10^-5(120))

Lf = 1.35324389 m

Lf = Li(1 + T)

Lf = 1.35(1 + 2*10^-5*120)

Lf = 1.35324 m

error caused by approximation = 1.35324389 - 1.35324 = 3.89*10^-6 m

c) Lf = 1.35*e^(0.02*120 ) = 14.88

Lf' = 1.35(1 + 0.02*120)

= 4.59 m

error = 14.88 - 4.59 = 10.29m

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