Question

The coefficient of linear expansion for some common materials are listed in the table to the right. You would like to design an emergency overheating alarm using one of these materials. In your design, at room temperature (23.0°C), a 6.26 cm long cylinder of the material will sit with one end against a wall and the other just barely touching a button. Once the heat causes the cylinder to lengthen by 0.0243 cm, the button will be activated, turning on the alarm. At what temperature will the alarm be triggered if the cylinder is made out of aluminum? At what temperature will the alarm be triggered if the cylinder is made out of titanium?

Answer 1

Coefficient of linear expansion of aluminium = _a = 24 x 10^-6  oC^-1

Coefficient of linear expansion of titanium = _t = 8.5 x 10^-6  oC^-1

Initial length of material = L = 6.26 cm

Expansion of material = L = 0.0243 cm

Room temperature = T_r = 23 ^oC

Temperature at which alarm is triggered is cylinder is made of aluminium = T_a

Temperature at which alarm is triggered is cylinder is made of titanium = T_t

L = TL

For aluminium,

L = _a(T_a - T_r)L

0.0243 = (24x10^-6)(T_a - 23)(6.26)

T_a - 23 = 161.74

T_a = 184.74 ^oC

For titanium,

L = _t(T_t - T_r)L

0.0243 = (8.5x10^-6)(T_t - 23)(6.26)

T_t - 23 = 456.68

T_t = 479.68 ^oC

If the cylinder is made of aluminium the alarm will be triggered at 184.74 ^oC.

If the cylinder is made of titanium the alarm will be triggered at 479.68 ^oC.