Incline Plane W/ Spring

grabcad

There are tons of variables in spring problems. The ways to consolidate all of these factors is through the law of conservation of energy:1/2mv_1^2+1/2kx_1^2+mgy_1 = 1/2mv_2^2+1/2kx_2^2+mgy_2+FLwhere m is the mass of the mass in question, v_1 and v_2 are the velocities of the mass at any two points, k is the spring constant, x_1 and x_2 are the displacement of the spring from its equilibrium position, y_1 and y_2 are the heights of the mass at any two points, F is the force of kinetic friction acting on the mass, and L is the distance traveled by the mass.Even though this equation looks daunting, it's easy to simplify if you know which points to examine.The key parts of this scenario occur when the spring is stretched to its maximum point and when the spring is at its equilibrium point.When the spring is stretched to its maximum point, the velocity of the mass is zero, and when the spring is at its equilibrium point x is zero and v is its maximum point. With these simplifications, you can choose them as your two points and set up a new equation:1/2kx_max^2+mgy_1 = 1/2mv_max^2+mgy_2+FLF is kinetic friction, so we can rewrite it to be:F = μNwhere μ is the coefficient of friction and N is the magnitude of the normal force.If we define the y-direction to be perpendicular to the surface of the incline plane and set up Newton's Second Law for this direction, then:N-mgcosθ = 0 --> N = mgcosθTherefore, F can be written as:F = μmgcosθIf we divide through by m and multiply through by 2, we get:k/mx_max^2 -v_max^2 = g(y_2-y_1) +μgcosθThis is a general formula for mass attached to a spring, and you would require an additional relation from the problem at hand to solve x_max or v_max.NOTE: Attached is a Linear Displacement VS Time graph. It damps over time, but does anybody know why it doesn't oscillate perfectly? It's a computer simulation, so I would assume it would be a perfect curve. Any help on this subject manner would be appreciated.