A piston of cross-section area A is fitted in a cylinder in which gas of volume V at pressure p is enclosed. It the gas obeys Boyle's law, then what is the angular frequency, if piston is displaced slightly ?

Options

A. `sqrt((Ag)/(V))`
B. `2sqrt((Ag)/(V))`
C. `sqrt((2Ag)/(V))`
D. `(3Ag)/(V)`

Answer

A

Explanation

The bulk modulus for gas is given by
`B=-(Delta p)/(Delta V//V)`
But the gas obeys Boyle's law, so pV=constant (isothermal process). In isothermal process, isothermal bulk modulus of gas is equal to the pressure of the gas at that instant of time
`Or B=p`
`therefore p= -(Delta p)/(DeltaV//V) rArr Delta p=-(p)/(V)Delta V`
`rArr (F)/(A)= -(p)/(V)Ax rArr F= -(pA^(2))/(V)x`
This equation is similar to
`F=-kx`
where, k = force constant of spring
So, `k=(pA^(2))/(V)`
Hence, angular frequency
`omega=sqrt((k)/(m))=sqrt((pA^(2))/(mV))=sqrt(((mg)A)/(mV)) " " (because p=(mg)/(A))`
`omega =sqrt((Ag)/(V))`