A block of mass m rests on a slope which is inclined at an angle $\backslash$theta to the horizontal. A massless spring with spring constant k is attached between the top of the slope and the block, parallel to the slope. Let g be the magnitude of the acceleration due to gravity, and let $\backslash$mu be the coefficient of friction between the block and the slope.

a$.$ Draw a diagram of the situation described, and label the forces acting on the block.

b$.$ Let x$($t$)$ be the extension of the spring at time t$.$ Using Newton$$s second law, show that x $=$ c$\backslash$omega $2$x while

the block is in motion, for constants c and $\backslash$omega $>0$ to be determined.

c$.$ Using the substitution y $=$ x $+$ c $,$ find a differential equation for y in terms of m and k$.$

d$.$ By writing down the general solution to the equation for y$,$ find x at time t$.$ Remember to define any constants that you introduce.