Question

The band in the figure below is stainless steel (coefficient of linear expansion = 17.3 ✕ 10⁻⁶°C⁻¹; Young's modulus = 18 ✕ 10¹⁰ N/m²). It is essentially circular with an initial mean radius of 4.6 mm, a height of 3.8 mm, and a thickness of 0.46 mm. If the band just fits snugly over the tooth when heated to a temperature of 76°C, what is the tension in the band when it cools to a temperature of 37°C?
N=

An illustration shows a horizontal band wrapped around the sides of a tooth.

Answer 1

a.)

Since Young's modulus is given by,

Y = (T/A)/(dL/L)

here,

Y = young's modulus of stainless steel = 18.0*10^10 N/m^2

T = tension in band = ??

A = area of band = Height*thickness = 4.6 mm*0.46 mm = 2.116 mm^2 = 2.116*10^-6 m^2

L = Initial length of band

dL = change in length due to change in temperature = *L*dT

here, = Coefficient of thermal expansion = 17.3*10^-6 per degC

dT = change in temperature = 37.0 - (76.0) = -39.0 degC (-ve sign means band will contract, so there will be tensile stress)

then, T = Y*A*dL/L = Y*A*(*L*dT)/L = Y*A**dT

T = 18*10^10*2.116*10^-6*17.3*10^-6*39.0 = 256.98 N

In two significant figures

T = Tension in the band = 260 N = 2.6*10^2 N

Let me know if you've any query.