The band on the tooth shown in the figure is stainless steel (coefficient of lin

Question

The band on the tooth shown in the figure is stainless steel (coefficient of linear expansion = 17.3 × 106 °C1; Young’s modulus = 18.0 × 1010 N/m2). It is essentially circular with an initial mean radius of 4.50 mm, a height of 3.75 mm, and a thickness of 0.550 mm. If the band just fits snugly over the tooth when heated to a temperature of 80.0°C, what is the tension in the band when it cools to a temperature of 37.0°C? Hint: Calculate the radius of the 37.0°C band if it were not on the tooth, and note that the tooth prevents the band from changing size as it cools.

Explanation / Answer

L = 2*phi*r = 2*pi*(0.0045) = 0.028

A = 0.000550*0.00375 = 2.06*10-6

DeltaL/L = alpha*DeltaT

DeltaL= L*alpha*T

= 0.028*17.3 × 106*(90-37) = 2.56*10-5

E = (T/A)/(DeltaL/L)

T = E*DeltaL*A/L

T = 18.0 × 1010*2.56*10-5 *2.06*10-6/( 0.028 ) = 339N

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