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Question: A steel ball initially at a pressure of \[1.0\times {{10}^{3}}Pa\] is heated from \({{20}^{0}}C\) to...

A steel ball initially at a pressure of 1.0×103Pa1.0\times {{10}^{3}}Pa is heated from 200C{{20}^{0}}C to 1200C{{120}^{0}}C keeping its volume constant. Find the pressure inside the ball. Coefficient of linear expansion of steel =12×10−6C12\times {{10}^{-6}}C and bulk modulus of steel=1.6×1011Nm−21.6\times {{10}^{11}}N{{m}^{-2}}.

Explanation

Solution

Stress is defined as the force per unit area applied to a material. The maximum stress of material can stand before it breaks or ultimate tensile stress. Tensile means the material is under tension. Stress is a physical quantity and force acting on the material is trying to stretch the body. Unit of pressure is Pascal’s (Pa).

Complete step-by-step solution:
Pressure is defined as the force exerted on an object. The formula of pressure is force per unit area. Different types of pressure are absolute, atmospheric, differential, and gauge pressure. Bulk modulus is defined as the ratio of infinitesimal pressure increase to the resulting relative decrease of the volume.
When the ball tries to expand its volume but its volume remains constant.
Then the force is given by:
P=−BΔVVP=\dfrac{-B}{\dfrac{\Delta V}{V}} ⋯⋯(1)\cdots \cdots (1)
Thermal expansion is given as below
V1=V(1+3αΔT) V1=V+3VαΔT \begin{aligned} & {{V}^{1}}=V(1+3\alpha \Delta T) \\\ & {{V}^{1}}=V+3V\alpha \Delta T \\\ \end{aligned}
Where ΔV=V1−V\Delta V={{V}^{1}}-V
ΔVV=3αΔT\dfrac{\Delta V}{V}=3\alpha \Delta T ⋯⋯(2)\cdots \cdots (2)
Substituting equation (2) in equation (1)
P=B(3αΔT)P=B(3\alpha \Delta T) ⋯⋯(3)\cdots \cdots (3)
From the data we know the values of ΔT=(120−20)\Delta T=(120-20) B=1.6×1011Nm−2B=1.6\times {{10}^{11}}N{{m}^{-2}} and α=12×10−6C\alpha =12\times {{10}^{-6}}C

& p=3\times 1.6\times {{10}^{11}}\times 12\times {{10}^{-6}}\times (120-20) \\\ & p=5.76\times 1{{0}^{8}}Pa \\\ \end{aligned}$$ Pressure inside the ball is $5.76\times {{10}^{8}}Pa$ **Note:** Students the reciprocal of the bulk modulus at temperature is called the isothermal compressibility bulk modulus is meaningful for fluids .Thermal conductivity is a measure of its ability to conduct heat. Conductivity is measured in watts per meter–kelvin. Inverse of thermal conductivity is thermal resistivity.