Expansion of Solid

Expansion of Solids:

• Increase in dimension of solid on heating,
• Linear Expansion: The expansion in length of an object.
• Superficial Expansion: The expansion in length and breadth of an object.
• Cubical Expansion: The expansion in three dimension.

Linear Expansion or One dimension expansion:

• Coefficient of linear expansion or linear expansivity{$\alpha$) is the change in length per unit original length per degree change in temperature. It's value is constant for a matter.
• $\alpha =\frac{\Delta l}{l\Delta \theta }$ or $l_{\theta }=l_o(1+\alpha \Delta \theta )$

Note:

Pendulum Clock

• The time period of a pendulum clock is given by $T=2\pi \sqrt{\frac{l}{g}}$
• When a pendulum clock keep correct time then time period of it must be 2 sec.
• During summer season, temperature increases so the length of pendulum clock also increase and it lose time or becomes late.
• During winter length of pendulum clock decrease so it gain time or becomes fast.
• To keep correct time at all temperature pendulum clock is made by invar, since value of linear expansivity for it is least
• Change in time per sec =$\frac{1}{2}\alpha \Delta \theta$

Differential expansion:

• The difference in expansions of different substances when they are heated through the same range of temperature is called the differential expansion.

When,

$d_2-d_1=(l_1\alpha -l_1^{\prime }{\alpha }^{\prime })\Delta \theta$

If the length of material chosen that then

                                       $l_1\alpha -l_1^{\prime }{\alpha }^{\prime }=0$

Force due to expansion or contraction:

• When a rod placed between two rigid supports is not free to expand on heating and not free to contract on cooling then the force or tension is developed along it.
• Force(F) = YA$\alpha \Delta \theta$ where Y is young's modulus of rigidity.
• Thermal Stress $(\frac{F}{A})=Y\alpha \Delta \theta$
• Thermal Strain $(\frac{\Delta l}{l})=\alpha \Delta \theta$
• Energy Density $(\frac{E}{V})=\frac{1}{2}stress\times strain=\frac{1}{2}Y(\alpha \Delta \theta {)}^2$

Bimetallic Thermostat:

• When two different metal strip are welded together then on heating, it bends towards one of lower linear expansivity.
• It is used in automatic electric switch to control temperature. Most popularly in heat sensor.
• Metal of higher linear expansivity lies on convex side and lower lines on concave side
• Radius of curvature (R)=$\frac{d}{({\alpha }_2-{\alpha }_1)\Delta \theta }$

Superficial Expansion or Two dimension expansion:

• Coefficient of superficial expansion or superficial expansivity($\beta$) is the change in area per unit original area per degree change in temperature . It's value is constant for a matter.
• $\beta =\frac{\Delta A}{A\Delta \theta }$ or $A_{\theta }=A_o(1+\beta \Delta \theta )$

Three dimensional Expansion or Cubical Expansion:

• Coefficient of cubical expansion or cubical expansivity($\gamma$) is the change in volume per unit original volume per degree change in temperature . It's value is constant for a matter.
• $\gamma =\frac{\Delta V}{V\Delta \theta }$ or $V_{\theta }=V_o(1+\gamma \Delta \theta )$

Relation between $\alpha ,\beta ,\gamma$

For an isotropic body

$\alpha =\frac{\beta }{2}=\frac{\gamma }{3}$

$\therefore \alpha :\beta :\gamma =1:2:3$

For anisotropic body

$\beta ={\alpha }_1+{\alpha }_2$ and $\gamma ={\alpha }_1+{\alpha }_2+{\alpha }_3$