Question

(a) Suppose a meter stick made of steel and one made of invar are the same length at 0°C. What is their difference in length at 48.0°C? mm (b) Repeat the calculation for two 34.0 m long surveyor's tapes. mm

Answer 1

Explanation:

Given that,

Initial temperature $T_{i}=0^{\circ}C$

Final temperature $T_{f}=48.0^{\circ}C$

Length = 34.0 m

We know that,

Coefficient of linear expansion of Invar is

$\alpha=0.9\times10^{-6}/^{\circ}C$

Coefficient of linear expansion of steel is

$\alpha=12\times10^{-6}/^{\circ}C$

We need to calculate the temperature difference is

$\Delta T=T_{f}-T_{i}$

$\Delta T=48.0-0$

$\Delta T=48.0^{\circ}C$

We need to calculate the change in length of Invar

Using formula of change in length

$\Delta l=l\times\alpha\times\Delta t$

Put the value into the formula

$\Delta l=34\times0.9\times10^{-6}\times48.0$

$\Delta l=0.0014688=1.47\times10^{-3}\ m$

$\Delta l= 1.47\ mm$

We need to calculate the change in length of steel

$\Delta l'=34\times12\times10^{-6}\times48.0$

$\Delta l'=0.019584=19.59\times10^{-3}\ m$

$\Delta l'=19.59\ mm$

Difference in length of two 34.0 m

Long surveyor's tapes is

$\Delta L=\Delta l'-\Delta l$

Put the value into the formula

$\Delta L=19.59-1.47=18.12\ mm$

The difference in length is 18.12 mm.

Hence, This is the required solution.