Problem $3$: Pushing a Box over a Rough Floor

You keep pushing the same $25kg$ box with a force vec$(F{)}_{\text{p}\text{u}\text{s}\text{h}}$ of magnitude of $82N$ at

an angle of $23$ below the horizontal, this time across a rough floor. Your friend

helps you tying a rope to the box and pulling at an angle of $37$ above the

horizontal, the tension on the pulling string has a magnitude of $65N.$

The friction from the floor given these conditions has a magnitude $|vec(F{)}_k|=36N.$

A$.$ On the diagram, draw a free$-$body diagram for the box. Make sure to label all the

forces on the box in your FBD$.$ The lengths of the vectors representing each force

should accurately represent the direction and relative strength of the forces.

B$.$ Determine the components $F_{T,x}$ and $F_{T,y}$ of the pulling force, $F_{\text{p}\text{u}\text{s}\text{h}\text{,}x}$ and

$F_{\text{p}\text{u}\text{s}\text{h}\text{,}y}$ of the pushing force and $F_{k,x}$ and $F_{k,y}$ of the kinetic friction.

C$.$ Argue about which Newton law $(1^{st}$ or $2^{nd})$ should be used to obtain information about the normal

force exerted on the box by the floor. Use it to calculate the normal force.

D$.$ Argue about which Newton law $(1^{st}$ or $2^{nd})$ should be used to obtain information about the horizontal

acceleration of the box. Determine the magnitude and direction of the acceleration of the box.