Question

A bimetal strip can be used as a thermometer and thermostat, a device that converts small changes in temperature to mechanical motion (see figure A below). A thermostat can be constructed by joining two pieces of metal, brass and steel, along their length as shown in figure B below. The brass and steel are both 60 cm long and 0.12 mm thick. If this bimetal strip is heated, the brass will expand more than the steel, causing the strip to bend. What change in temperature is needed to make the bimetal strip curl to a full circle as shown in figure C below?

Answer 1

Given

Lenth of one strip L = 0.60 m

Thickness of one strip t = 0.12 x 10^-3 m

Known

Linear thermal expansionn coefficient of brass α_B = 19 x 10^-6 K^-1

Linear thermal expansionn coefficient of brass α_s = 13 x 10^-6 K^-1

Solution

The Difference in radius r₁ - r₂ = t = 0.12 x 10^-3 m

The new length of the srips are equal to the circumference of the circles.

Circumference of brass circle S_B = 2πr₁

Circumference of steel circle S_S = 2πr₂

Change in length of the brass sripe

ΔL = α_BLΔT

2πr₁ - L = α_BLΔT ….(1)

Change in length of the steel sripe

ΔL = α_sLΔT

2πr₂- L = α_sLΔT   ….(2)

1. - (2)

2πr₁ -2πr₂ = (α_B -α_s )LΔT

2π(r₁ -r₂ ) = (α_B -α_s )LΔT

2πt = (α_B -α_s )LΔT

ΔT = 2πt/(α_B -α_s )L

ΔT = 2 x 3.14 x 0.12 x 10^-3 / ( 19 x 10^-6- 13 x 10^-6) x 0.60

ΔT = 209 ^OC

An increase in temperature by 209 ^oC would form a full circle