Instead of using a coil a gas spring uses the properties of

Instead of using a coil, a gas spring uses the properties of a gas in a cylinder to obtain the behaviour of a spring. The diagram below shows a gas spring of length x. Note that F_spring = F_s = 0 when the spring is at its \"natural length\" of x = x_0. The force due to the gas spring is determined by the pressure of the gas within it. Since the temperature of the gas remains constant, and no gas escapes, the pressure is determined by Boyle s Law, p_1 V_1 = p_2 V_2 = k where p is pressure and V is volume. The force due to the gas is F = pA where A is the cross-sectional area of the surface the pressure, and thus the force, applies to. Note that the air outside the gas spring will also exert a force, but that the pressure for the air will remain constant. Determine the resting length of the spring, x_infinity, when the mass is attached. Using MATLAB\'s ode4 5 function, solve the ODE if m = 0.13, c = 0.27, k = 0.35, g = 9.81, and x_0 = 0.1, with x(0) = 0.03 and x\'(0) = 0, for 0 lessthanorequalto t lessthanorequalto 5. Plot your solution. Write an algorithm for the process of linearising an ODE. Apply this to linearise the ODE (around x = x_infinity). Using the same values from part (d), solve the linearised ODE you obtained in part (e) using ode4 5. Plot this solution on the same axes as your solution to part (d), and comment on the result.

Solution

Granularity (also called \"graininess\", the quality of being grainy) is the extent to which a material or system is composed of distinguishable pieces or grains. It can either refer to the extent to which a larger entity is subdivided, or the extent to which groups of smaller indistinguishable entities have joined together to become larger distinguishable entities. For example, a kilometer broken into centimeters has finer granularity than a kilometer broken into meters. In contrast, molecules of photographic emulsion may clump together to form distinct noticeable granules, reflecting coarser granularity.

Coarse-grained materials or systems have fewer, larger discrete components than fine-grained materials or systems. A coarse-grained description of a system regards large subcomponents while a fine-grained description regards smaller components of which the larger ones are composed.

The concepts granularity, coarseness, and fineness are relative, used when comparing systems or descriptions of systems. An example of increasingly fine granularity: a list of nations in the United Nations, a list of all states/provinces in those nations, a list of all cities in those states, etc.

The terms fine and coarse are used consistently across fields, but the term granularity itself is not. For example, in investing, more granularity refers to more positions of smaller size, while photographic film that is more granular has fewer and larger chemical \"grains.\" Similarly, sugar that is more granular has fewer and larger gr