# A large steel wheel is to be fitted on to a shaft of the same material. At 29^{\circ}C, the outer...

## Question:

A large steel wheel is to be fitted on to a shaft of the same material. At 29{eq}^{\circ} {/eq}C, the outer diameter of the shaft is 11.70 cm and the diameter of the central hole in the wheel is 9.69 cm. The shaft is cooled using dry ice. At what temperature of the shaft does the wheel slip on the shaft? Assume the coefficient of linear expansion of the steel to be constant over the required temperature range: {eq}\alpha_{steel} = 1.20 \times 10^{-5} K^{-1} {/eq}.

## Thermal expansion:

The effect of temperature change, which causes elongation or contraction in a body, is known as thermal expansion. There are three types of thermal expansion such as,

• Linear expansion
• Superficial expansion
• Cubical expansion

The standard unit of thermal expansion is watt per meter kelvin

Given Data:

• The outer diameter of the shaft is, {eq}{d_1} = 11.70\;{\rm{cm}} {/eq}
• The diameter of the central hole in the wheel is, {eq}{d_2} = 9.69\;{\rm{cm}} {/eq}
• The coefficient of linear expansion of the steel is, {eq}{\alpha _{{\rm{steel}}}} = 1.20 \times {10^{ - 5}}\;{{\rm{K}}^{ - 1}} {/eq}
• The temperature is, {eq}T = 29^\circ {\rm{C}} = 302\;{\rm{K}} {/eq}

The shaft is cooled using dry ice and the temperature becomes {eq}{T_1} {/eq},

For the calculation of temperature {eq}{T_1} {/eq} use the relation,

{eq}\Delta d = {d_1}{\alpha _{{\rm{steel}}}}\left( {{T_1} - T} \right) {/eq}..............(1)

The change in diameter after the wheel will slip on the shaft is calculated as,

{eq}\begin{align*} \Delta d &= 9.69 - 11.70\\ \Delta d &= - 2.01\;{\rm{cm}} \end{align*} {/eq}

Substitute the values in equation (1)

{eq}\begin{align*} - 2.01 &= 11.70 \times 1.20 \times {10^{ - 5}}\left( {302 - {T_1}} \right)\\ {T_1} &= 14.01 \times {10^3}\;{\rm{K}}\\ {T_1} &= 13.73 \times {10^3}^\circ {\rm{C}} \end{align*} {/eq}

Thus, the wheel will slip on the shaft when the temperature of the shaft is {eq}13.73 \times {10^3}^\circ {\rm{C}} {/eq}.