Question

Can you solve this problem in Mechanics?

Answer 1

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

Explanation:

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.