A gas spring is a spring mechanism that rather than using a

A gas spring is a spring mechanism that, rather than using a physical coil to generate the spring behavior, uses the properties of a gas to achieve this behavior. Here is a simplified representation of a gas spring: The force exerted by the gas spring is dictated by the pressure of the gas within it. As the temperature of the gas remains constant, and no gas escapes, the pressure is dictated by Boyle\'s Law, P_1V_1 = P_2V_2 = k where p is pressure and V is volume. The force due to the gas is given by F = PA where A is the area of the surface the pressure, and thus the force, applies to (in our case, this area is the cross sectional area).

Solution

a) From FBD
mg + F*x - cv = ma
mg + F - cx\' = mx\"
now, F = pA
At some displacement x
p*x*A = k ( constant)
so, mg + pA - cx\' = mx\"
mg + k/x - cx\' = mx\"
using initial conditions
mx\" + cx\' - k(1/x - 1/xo) = mg ( where x0 is the initial position of the mass)

b) for all forces balanced, x\" = 0
cx\' - k(1/x - 1/xo) = mg
cx\'= mg - k/xo + k/x
integrating
cx = mgt - kt/xo + kln(x) [ from x = xo to x = xinf]
here xinf is the length required