A gas spring is a spring mechanism that, rather than using a physical coil to ge

Question

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). a) (*) By considering the forces involved - the force due to pressure in the gas spring and the force due to gravity, as well as air resistance (linear) - show that the motion of mass m is described by m d^2x/dt^2 + cdx/dt - k/x + kx0 = mg b) (*) Determine the resting length of the spring, x_infinity, when the mass is attached. That is, the length for which all forces balance.

Explanation / Answer

a)    Here, Differential equation for damping motion

=> d2x/dt2 + (c/m)dx/dt + (k/m)x = 0

Here, value of (k/m)x = (k/mxo) - (k/mx)

=> equation becomes

=>    d2x/dt2 + (c/m)dx/dt + (k/mxo) - (k/mx) = 0

This is equal to the value of gravity , g

=>    d2x/dt2 + (c/m)dx/dt + (k/mxo) - (k/mx) =   g

=>    md2x/dt2 + (c)dx/dt + (k/xo) - (k/x) =   mg

b)    Resting length =   (1/xo) - (1/x)

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