Question

Consider a steel tape measure with cross-sectional area, A = 0.0625 inches squared, and length L = 3, 600 inches at room temperature. How much error will occur if this tape measure is used on a hot day? Assume it is 130F and the coefficient of thermal expansion is α = 5× 10 −6 1/F. Does the error depend on the distance being measured?

Answer 1

Answer:

e = -0.00031 ( the -ve sign is due to the increase in length)

The error depends on the distance measured

Explanation:

Cross Sectional Area of the tape, A = 0.0625 in²

Length of the steel tape, L = 3600 in

Normal room temperature, T₁ = 68°F

Temperature of the hot day, T₂ = 130°F

ΔT = T₂  - T₁ = 130 - 68

ΔT = 62°F

Coefficient of Linear expansion $\alpha = 5 * 10^{-6} F^{-1}$

The coefficient of linear expansion is given by the formula:

$\alpha = \frac{\triangle L}{L* \triangle T} \\\triangle L = \alpha L \triangle T\\\triangle L = 5* 10^{-6} * 3.6* 10^3 * 62\\\triangle L = 1.11 6 in$

Since the length is increased, the error will be given by the formula:

$e = \frac{-\triangle L}{L} \\\\e = \frac{-1.116}{3600}$

e = -0.00031 ( the -ve sign is due to the increase in length)

Since the error is a function of length and change in length, it depends on the distance measured