A massless piston which can slide inside a smooth vertical cylinder is attached with a light spring initially in its natural length as shown with force constant k. Now, a mass 'm' is placed on the piston. 'A' is cross-sectional area of the cylinder and the piston. Find the work done by the gas as it expands, if the piston moves up by a distance x (consider P_0 as the atmospheric pressure).

A massless piston which can slide inside a smooth vertical cylinder is

1.	A massless piston which can slide inside a smooth vertical cylinder is attached with a light spring initially in its natural length as shown with force constant k. Now, a mass 'm' is placed on the piston. 'A' is cross-sectional area of the cylinder and the piston. Find the work done by the gas as it expands, if the piston moves up by a distance x (consider P_0 as the atmospheric pressure).
Answer» Solution :As the PISTON is massless, net force on it at every instant is zero. `=`upward force = downward force `=PA = kx +mg + P_0A` `(or)P=P_0+(kx) /(A) +(mg)/(A ) ,dW =PDV =P( Adx) = (AP_0+kx +mg) dx ` ` thereforeW= int_(0)^(x)PdV(or)W= int_(0)^(x) (AP_0+kx +mg) dx=P_0Ax+1/2kx^2+mgx` (as ` Ax = Delta V)` `(or)W=P_0 DeltaV +1/2kx^2 +mgx` The result can be STATED in a DIFFERENT manner as under. The gas does work against the atmospheric pressure `P_0` (which is constant), the spring force kx (which varies linearly with x) and the gravity force mg (which is again constant). ` thereforeW_1= `Work done against `P_0 =P_0 Delta V` `W_2= `work done against `kx = 1/2kx^2` and `W_3` = Work done against mg = mgx So, that `W_("TOTAL") = W_1+W_2+W_3 =P_0Delta V +1/2 kx^2+ mgx`