3A. A 100-N block on a rough 30degrees-incline is acted upon by a force P as shown causing it to accelerate at 2 m/s^2 up the plane. The coefficient of kinetic friction is 0.30.
(a) Draw the FBD of the block
(b) Find the magnitude of P
$\mathrm{Solution}:-\mathrm{Given}\mathrm{that}\phantom{\rule{0ex}{0ex}}\mathrm{W}=100\mathrm{N}\phantom{\rule{0ex}{0ex}}\mathrm{a}=2\mathrm{m}/{\mathrm{s}}^{2}\phantom{\rule{0ex}{0ex}}{\mathrm{\mu}}_{\mathrm{k}}=0.3$
$\left(\mathrm{a}\right)\mathrm{FBD}\mathrm{of}\mathrm{block}\mathrm{is}\mathrm{shown}\mathrm{below}:$
$\left(\mathrm{b}\right)\mathrm{Now},\mathrm{applying}\mathrm{Newton}\text{'}\mathrm{s}\mathrm{first}\mathrm{law}\mathrm{in}\mathrm{y}\mathrm{direction};\phantom{\rule{0ex}{0ex}}\sum _{}{\mathrm{F}}_{\mathrm{y}}=0\phantom{\rule{0ex}{0ex}}\mathrm{N}-\mathrm{W}\mathrm{Cos}30-\mathrm{PSin}50=0\phantom{\rule{0ex}{0ex}}\mathrm{N}=\mathrm{W}\mathrm{Cos}30+\mathrm{PSin}50-\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{Again},\mathrm{applying}\mathrm{Newton}\text{'}\mathrm{s}\mathrm{second}\mathrm{law}\mathrm{in}\mathrm{x}\mathrm{direction};\phantom{\rule{0ex}{0ex}}\sum _{}{\mathrm{F}}_{\mathrm{x}}=\mathrm{ma}\phantom{\rule{0ex}{0ex}}\mathrm{P}\mathrm{Cos}50-{\mathrm{f}}_{\mathrm{k}}-\mathrm{WSin}30=\frac{\mathrm{W}}{\mathrm{g}}\mathrm{a}\phantom{\rule{0ex}{0ex}}\mathrm{where}{\mathrm{f}}_{\mathrm{k}}=\mathrm{kinetic}\mathrm{friction}=\mathrm{\mu N}=\mathrm{\mu}(\mathrm{W}\mathrm{Cos}30+\mathrm{PSin}50)\phantom{\rule{0ex}{0ex}}\mathrm{P}\mathrm{Cos}50-\mathrm{\mu}(\mathrm{W}\mathrm{Cos}30+\mathrm{PSin}50)-\mathrm{WSin}30=\frac{\mathrm{W}}{\mathrm{g}}\mathrm{a}\phantom{\rule{0ex}{0ex}}\mathrm{P}\mathrm{Cos}50-0.3(100\mathrm{Cos}30+\mathrm{PSin}50)-100\mathrm{Sin}30=\frac{100}{9.81}\times 2\phantom{\rule{0ex}{0ex}}\mathrm{P}\mathrm{Cos}50-0.3\mathrm{PSin}50=\frac{100}{9.81}\times 2+100\mathrm{Sin}30+0.3\times 100\mathrm{Cos}30\phantom{\rule{0ex}{0ex}}\mathrm{P}=\frac{\frac{100}{9.81}\times 2+100\mathrm{Sin}30+0.3\times 100\mathrm{Cos}30}{\mathrm{Cos}50-0.3\mathrm{Sin}50}\phantom{\rule{0ex}{0ex}}\mathbf{P}\mathbf{=}\mathbf{233}\mathbf{.}\mathbf{35}\mathbf{}\mathbf{N}$