$2.)$ Observe the situation below where a mass $(200$ N weight$)$ is at rest on an inclined surface $($see the diagram$).$ The mass is placed at some point along the line CD$.$ Assume that a frictional force is acting $($along the direction DC$)$ to prevent it from sliding down.

a$)$ Calculate $2$ vectors that lie ON the inclined plane.

b$)$ Calculate the unit normal vector to the inclined plane.

c$)$ Calculate the unit vector in the direction of the static friction force.

d$)$ Draw a CLEAR LARGE Free$-$Body$-$Diagram of all forces that act on the disk. Clearly label all forces & moments that act on the mass

$(2$ pts$)$

$(5$ pts$)$

$(2$ pts$)$

$(3$ pts$)$

e$)$ Write down a vector equation for equilibrium of all Forces on the mass $($in terms of $i,j,k).$ As your unknowns use the friction vector $($assumed to act along the line CD$)$ and the normal force vector. Calculate all unit vectors required.

$(4$ pts$)$

f$)$ Write this vector equation as $3$ scalar equations for each of the $3$ dimensions. There will only be $2$ unknowns. Finally represent them as a matrix equation $Ax=B."A"$ is the matrix of coefficients, $"x"$ is the column matrix of unknowns, $"$B$"$ is the column matrix of constants.

$(5$ pts$)$

g$)$ Solve the equations, for the friction force and the normal force. You may express the solutions as scalars.

h$)$ What is the coefficient of static friction? Show your work.

$(3$ pts$)$

$(3$ pts$)$

i$)$ Solve for the same unknown forces using $2$D Free Body Diagram and $2$D analysis to solve for the $2$ unknowns you originally had.

$(3$ pts$)$

$30$ POINTS