Can you solve this problem in Mechanics?

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Answer 1 · 2016-11-16T17:15:21

$x(t) = (x_e-(gm)/k tan(a))(cos(sqrt(k/m)t)-1)$

Considering the movement projected over the ramp, applying Newton's second law and making $alpha = dy/dx=a$

$-m g sin(alpha)-k(x-x_e)cos(alpha)=m ddotx cos(alpha)$ or

$ddot x+k/m x+g tan(alpha)-kx_ecos(alpha)=0$

This second order linear differential equation has the general solution.

$x(t)=x_e+C_1 Cos(sqrt[k/m] t) + C_2 sin(sqrt[k/m] t) - ( g m tan(alpha))/k$

$C_1,C_2$ are determined according to the initial conditions:

${(x(0)=x_0),(dot x(0)=0):}$

so finally

$x(t) = (x_e-(gm)/k tan(a))(cos(sqrt(k/m)t)-1)$

The $y(t)$ position formulation, is let as an exercise to the reader.