The free body diagram of the crate can be shown as,
According to free body diagram, the net force acting on the crate is,
[tex]F=mg\sin \theta-f\ldots\ldots\text{ (1)}[/tex]The frictional force acting on the crate is,
[tex]f=\mu N[/tex]The normal force acting on crate is,
[tex]N=mg\cos \theta[/tex]Therefore, the frictional force becomes,
[tex]f=\mu mg\cos \theta[/tex]According to Newton's second law of motion, the net force acting on the crate is,
[tex]F=ma[/tex]Therefore, equation (1) becomes,
[tex]\begin{gathered} ma=mg\sin \theta-\mu mg\cos \theta \\ a=g(\sin \theta-\mu\cos \theta) \end{gathered}[/tex]Substitute the known values,
[tex]\begin{gathered} a=(9.8m/s^2)(\sin 32.7^{\circ}-(0.33)\cos 32.7^{\circ}) \\ =(9.8m/s^2)(0.540-(0.33)(0.842)) \\ =(9.8m/s^2)(0.262) \\ \approx2.57m/s^2 \end{gathered}[/tex]Thus, the acceleration of the moving crate is 2.57 m/s2.
