A large steel wheel is to be fitted on to a shaft of the same material. At $27\,^{\circ} C ,$ the outer diameter of the shaft is $8.70\; cm$ and the diameter of the centrall hole in the wheel is $8.69 \;cm$. The shaft is cooled using 'dry ice'. At what temperature (in $^oC$) of the shaft does the wheel slip on the shaft? Assume coefficient of linear expansion of the steel to be constant over the required temperature range: $\alpha_{steel} =1.20 \times 10^{-3} \;K ^{-1}$
The given temperature, $T=27^{\circ} C$ can be written in Kelvin as
$27+273=300 K$
Outer diameter of the steel shaft at $T, d_{1}=8.70 cm$
Diameter of the central hole in the wheel at $T, d_{2}=8.69 cm$
Coefficient of linear expansion of steel, $\alpha$ steel $=1.20 \times 10^{-5} K ^{-1}$
After the shaft is cooled using "dry ice', its temperature becomes $T_{1}$.
The wheel will slip on the shaft, if the change in diameter, $\Delta d=8.69-8.70$
$=-0.01 cm$
Temperature $T_{1},$ can be calculated from the relation:
$\Delta d=d_{1} \alpha_{\text {steel }}\left(T_{1}-T\right)$
$=8.70 \times 1.20 \times 10^{-5}\left(T_{1}-300\right)$
$\left(T_{1}-300\right)=95.78$
$\therefore T_{1}=204.21 K$
$=204.21-273.16$
$=-68.95^{\circ} C$
Therefore, the wheel will slip on the shaft when the temperature of the shaft is $-69\,^{\circ} C$
We would like to make a vessel whose volume does not change with temperature (take a hint from the problem above). We can use brass and iron $\left( {{\beta _{{\text{v brass }}}} = 6 \times {{10}^{ - 5}}/K} \right.$ and $\left. {{\beta _{{\text{viron }}}} = 3.55 \times {{10}^{ - 5}}/K} \right)$ to create a volume of $100\,cc$ . How do you think you can achieve this.
The volume of the bulb of a mercury thermometer at $0^o C$ is $V_0$and cross section of the capillary is $A_0$. The coefficient of linear expansion of glass is $a_g$ $per ^o C$ and the cubical expansion of mercury $\gamma_m$ $per ^o C$. If the mercury just fills the bulb at $0^o C$, what is the length of mercury column in capillary at $T^o C.$
Given below are two statement : one is labelled as Assertion $A$ and the other is labelled as Reason $R$.
Assertion $A$ : When a rod lying freely is heated, no thermal stress is developed in it.
Reason $R :$ On heating the length of the rod increases.
In the light of the above statements, choose the correct answer from the options given below
The weight of sphere in air is $50\ g$. Its weight $40\ g$ in a liquid, at temperature $20\,^o C$. When temperature increases to $70\,^o C$ , it weight becomes $45\ g$, then the ratio of densities of liquid at given two temperature is
The coefficient of volume expansion of glycerin is $49 \times 10^{-5} \;K ^{-1} .$ What is the fractional change in its density for a $30\,^{\circ} C$ rise in temperature?