A brass rod of length $50\; cm$ and diameter $3.0 \;mm$ is jotned to a steel rod of the same length and diameter. What is the change in length of the combined rod at $250\,^{\circ} C ,$ if the original lengths are at $40.0\,^{\circ} C ?$ Is there a 'thermal stress' developed at the junction? The ends of the rod are free to expand (Co-efficient of linear expansion of brass $=2.0 \times 10^{-5} \;K ^{-1},$ steel $=1.2 \times 10^{-5}\; K ^{-1} J$
Initial temperature, $T_{1}=40^{\circ} C$
Final temperature, $T_{2}=250^{\circ} C$
Change in temperature, $\Delta T=T_{2}-T_{1}=210^{\circ} C$
Length of the brass rod at $T_{1}, l_{1}=50 cm$
Diameter of the brass rod at $T_{1}, d_{1}=3.0 mm$
Length of the steel rod at $T_{2}, l_{2}=50 cm$
Diameter of the steel rod at $T_{2}, d_{2}=3.0 mm$
Coefficient of linear expansion of brass, $\alpha_{1}=2.0 \times 10^{-5} K ^{-1}$
Coefficient of linear expansion of steel, $\alpha_{2}=1.2 \times 10^{-5} K ^{-1}$
For the expansion in the brass rod, we have:
$\frac{\text { Change in length }\left(\Delta I_{1}\right)}{\text { Original length }\left(I_{1}\right)}=\alpha_{1} \Delta T$
$\therefore \Delta l_{1}=50 \times\left(2.1 \times 10^{-5}\right) \times 210$
$=0.2205 cm$
For the expansion in the steel rod, we have:
$\frac{\text { Change in length }\left(\Delta l_{2}\right)}{\text { Original length }\left(l_{2}\right)}=\alpha_{2} \Delta T$
$\therefore \Delta l_{2}=50 \times\left(1.2 \times 10^{-5}\right) \times 210$
$=0.126 cm$
Total change in the lengths of brass and steel,
$\Delta l=\Delta l_{1}+\Delta l_{2}$
$=0.2205+0.126$
$=0.346 cm$
Total change in the length of the combined rod $=0.346\, cm$
Since the rod expands freely from both ends, no thermal stress is developed at the junction
A block of wood is floating on water at $0^{\circ} C$ with volume $V_0$ above water. When the temperature of water increases from $0$ to $10^{\circ} C$, the change in the volume of the block that is above water is best described schematically by the graph.
Write relation between coefficient of linear and volume expansion.
At $40\,^oC$, a brass wire of $1\, mm$ is hung from the ceiling. A small mass, $M$ is hung from the free end of the wire. When the wire is cooled down from $40\,^oC$ to $20\,^oC$ it regains its original length of $0.2\, m$. The value of $M$ is close to ........$kg$ (Coefficient of linear expansion and Young's modulus of brass are $10^{-5}/^oC$ and $10^{11}\, N/m^2$, respectively; $g = 10\, ms^{-2}$)
A brass wire $1.8\; m$ long at $27\,^{\circ} C$ is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of $-39\,^{\circ} C ,$ what is the tension developed in the wire, if its diameter is $2.0 \;mm$ ? Co-efficient of Itnear expansion of brass $=2.0 \times 10^{-5}\; K ^{-1} ;$ Young's modulus of brass $=0.91 \times 10^{11} \;Pa$
A one litre glass flask contains some mercury. It is found that at different temperatures the volume of air inside the flak remains the same. ...... $cc$ is the volume of mercury in this flask if coefficient of linear expansion of glass is $9 \times 10^{-6} /^o C$ while of volume expansion of mercury is $1.8 \times {10^4}\,/^\circ C$