An ideal gas enclosed in a vertical cylindrical container supports a freely moving pistion of mass M. The pistion and the cylinder have equal cross sectional area A. When the pistion is in equilibrium, the volume of the gas is V_(0) and its pressure is P_(0). The pistion is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the pistion executes a simple harmonic motion with frequency

An ideal gas enclosed in a vertical cylindrical container supports a f

1.	An ideal gas enclosed in a vertical cylindrical container supports a freely moving pistion of mass M. The pistion and the cylinder have equal cross sectional area A. When the pistion is in equilibrium, the volume of the gas is V_(0) and its pressure is P_(0). The pistion is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the pistion executes a simple harmonic motion with frequency
Answer» <P>`(1)/(2pi) (A gamma P_(0))/(V_(0) M)` `(1)/(2pi) (V_(0)MP_(0))/(A^(2) gamma)` `(1)/(2pi) sqrt((A^(2) gamma P_(0))/(MV_(0)))` `(1)/(2pi) sqrt((MV_(0))/(A gamma P_(0)))` Solution :`(Mg)/(A) = P_(0)` `Mg = P_(0) A` `P_(0) V_(0)^(gamma) = (P_(0) + Delta P_(0)) (V_(0) - Delta V_(0))^(gamma)` `= (P_(0) - gamma P_(0) (Delta V_(0))/(V_(0)) + Delta P_(0))` or `Delta P_(0) = gamma P_(0) (Delta V_(0))/(V_(0))` But `Delta V = Ax` where A is then area of cross - section of the PISTION `Delta P_(0) = (gamma P_(0) A)/(V_(0)) x` RESTORING force `F =- Delta P_(0) xx A =- (gamma P_(0) A^(2))/(V_(0))x` Compare above equation with `f_(res) =-kx` `f = (1)/(2pi) sqrt((gamma P_(0) A^(2))/(MV_(0)))`