Problem on Thermal Expansion

Question
A circular hole of diameter 2.00 cm is made in an aluminium plate at 0 ° C .what will be the diameter at 100° C?
α for aluminium = 2.3 * 10-3 / ° C

Solution
Thermal Expansion
- Increase in dimension of body due to increase in temperature is call thermal expansion
- Most of the solid material expand when heated

-Consider a rod of length L then for small change in temperature ΔT,the fractional change in length ΔL/L is directly propertional to ΔT
ΔL/L=αΔT --(2)
or ΔL=αLΔT --(3)
- constant α characterizes the thermal expansion properties of a particulaqr material and it is known as coefficient of linear expansion.
- for materials having no prefential direction,every linear dimension changes according to equation (3) and L could equally well represent the thickness of the rod,side lenght of the square sheet etc
-Normally metals expand more and have high value of α
- Agian consider the intial surface area A of any surface .Now when the temperature of the body is increases by ΔT ,the increase in surface area is given by
ΔA=αAAΔT ----(4)
where αA is the coefficient of area expansion
-Similary we can define coefficient of volume expansion as fractional change in volume ΔV/V of a substance for a temperature change ΔT

as ΔV=αVVΔT ----(5)
- K-1 is the unit of these coefficents expansions
- These three coefficent are not strictly constant for a substance and these value is depends on temperature range in which they are measured.

Relation between volume and linear coefficient of expansion for solid materail: Consider a solid parallopide with dimension L1,L2 and L3
then volume is
V= L1L2L3
when temperature increase by a amount ΔT then each linear dimension changes and then new volume is
V+ΔV=L1L2L3(1+αL )
V+ΔV=V(1+αLΔT)3
V+ΔV=V(1+3αLΔT+3αL2ΔT2L3ΔT3)
if ΔT is small the higher order can be neglected.Thus we find
V+ΔV=V(1+3αLΔT)
ΔV=3αLVΔT
Comparing this with equation (5) we find
αV=3αL

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