Question

Interpret and model using appropriate Mathem 5) Suppose a particle of mass m in a gravitational feld g is atached to a horizontal spring whose stifes is k and suppose the particle is allowed to slide on a horizontal surface such that pu is the kinetic coefficient of friction between the surface and the mass as it slides (a) Write a differential equation describing the motion of the mass on the surface when it is subject to only the spring's force and the friction force from the surface. (b) If the spring is originally compressed by an amount 6 and then released from rest, find the total distance the mass moves in its initial direction of motion before turning around. Express your answer in terms of the parameters given in the problem. anine İndicating which is which and give the

Answer 1

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Both frictional force and strong force off spring acts opposite to the direction motion of mass at any time So F^rightarrow = F_s + f = -k times mu_k mg So ma^rightarrow = md^2 x/dt^2 = -t x - mu_k mg \r\n Differential equation is d^2 x/dt^2 + k/m x + mu_k g = 0 \r\n b) using energy conservation principle change calculate potential energy of spring = change in kinetic energy of mass + work done against friction initial velocity of mass v_i = 0 m/s (at net) At turning point its velocity is also so the delta KE at tuning point is zero and all the potential energy goes into doing work against friction so 1/2 k s^2 = mu_k m g h h: dielectric move implies h = k s^2/2 my_k mg