Thermal Expansion Concept Page - 6

Example

Change in time period of a pendulum due to linear thermal expansion of rod

A pendulum clock runs fast by 5$5$ seconds per day at 20∘C${20}^{\circ }C$ and goes slow by 10$10$ seconds per day at 35∘C${35}^{\circ }C$. At what temperature does it show correct time? We know, α=dl(dt×l)$\alpha ={\displaystyle \frac{dl}{(dt\times l)}}$ [dl=$dl=$ length increase, dt=$dt=$ temperature increase]
or, dl=α×dt×l$dl=\alpha \times dt\times l$  ---- (1)
Now. T=2πlg−−√$T=2\pi \sqrt{{\displaystyle \frac{l}{g}}}$ ---(2) [T=$T=$ time period]
T+dT=2π(l+dl)g−−−−−−−√$T+dT=2\pi \sqrt{{\displaystyle \frac{(l+dl)}{g}}}$
or, T(1+dTT)=2πlg−−√(1+dll)−−−−−−−√$T(1+{\displaystyle \frac{dT}{T}})=2\pi \sqrt{{\displaystyle \frac{l}{g}}}\sqrt{(1+{\displaystyle \frac{dl}{l}})}$ ---- (3)
Comparing with (2), we get:
1+dTT=(1+dll)1/2=1+12dll$1+{\displaystyle \frac{dT}{T}}=(1+{\displaystyle \frac{dl}{l}}{)}^{1/2}=1+{\displaystyle \frac{1}{2}}{\displaystyle \frac{dl}{l}}$
or, dTT=12dll${\displaystyle \frac{dT}{T}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{dl}{l}}$
or, dT=12dll×T$dT={\displaystyle \frac{1}{2}}{\displaystyle \frac{dl}{l}}\times T$
Using (1), we get:
dT=12(α×dt×l×T)l$dT={\displaystyle \frac{1}{2}}{\displaystyle \frac{(\alpha \times dt\times l\times T)}{l}}$
or, dT=12(α×dt×T)$dT={\displaystyle \frac{1}{2}}(\alpha \times dt\times T)$
In case 1, the clock goes fast by 5$5$ sec at 200C${20}^{0}C$
In case 2, the clock goes slow by 10$10$ sec at 350C${35}^{0}C$
So,10=12α(35−t)×86400$10={\displaystyle \frac{1}{2}}\alpha (35-t)\times 86400$
and 5=12α(t−20)×86400$5={\displaystyle \frac{1}{2}}\alpha (t-20)\times 86400$
Solving these two equations we get, t=250C$t={25}^{0}C$

Definition

Thermal stress

If a body is allowed to deform freely on temperature change, no load or stress will be induced in the body. On the other hand where temperature deformation is not permitted, an internal stress is created inside the body termed as thermal stress. An example of thermal stresses is the tensile stresses produced in a wire stretched between fixed supports as the wire is cooled.

Definition

Example of thermal stress in construction design

Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes to account for the thermal stresses .

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