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An aluminium sphere at 20^(@)C is kept a...

An aluminium sphere at `20^(@)C` is kept at 1 standard atmosphere pressure, in a pressure chamber covered with oil. What should be the rise in pressure on the sphere to keep its volume unchanged even at `35^(@)C`? Coefficient of linear expansion of aluminium, `alpha=23 times 10^(6@)C^(-1)` and its bulk modulus of elasticity, `K=7.7 times 10^(10)N*m^(-2).`

Text Solution

Verified by Experts

Let the volume of the sphere be `V_(0)` at `20^(@)C ` and V at `35^(@)C.`
`therefore" " V=V_(0){1+3alpha(35-20)}=V_(0){1+3 times 23 times 10^(-6) times 15}`
or, `" " V/V_(0)=1+1035 times 10^(-6) " or, " V/V_(0)-1=1035 times 10^(-6)`
`therefore " " (V-V_(0))/V_(0)=1035 times 10^(-6)`
Let the required increase in pressure be p.
`therefore` Bulk modulus of elasticity, `k=(p/(V-V_(0)))/V_(0)`
or, `" "p=K*(V-V_(0))/V_(0)=7.7 times 10^(10) times 1035 times 10^(-6)N*m^(-2)`
`" " =(7.7 times 10^(4) times 1035)/101300` standard atmosphere
`" "` = 786.7 standard atmosphere.
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