Answer:
(a) The box will
remain at rest it force of friction \[(f)>mg\,\sin \theta \]
The
box will first start to slide down the plane if
\[mg\,\,\sin
\,\theta \,=f=\,R=\,mg\,\cos \theta \]
or \[\tan
\,\theta \,=\mu \] or
\[\theta \,=\,{{\tan }^{-1}}\,(\mu )\]
(b)
Force acting oil die box down the plane
\[=\,mg\,\sin
\,\alpha -\,f=\,mg\,\sin \alpha -\,\mu R\]
\[=\,mg\,\,\sin
\,\alpha -\,\mu \,mg\,\cos \alpha \]
\[=\,mg\,\,(\sin
\,\alpha -\,\mu \,\cos \alpha )\]
(c)
\[F=mg\text{ }\sin \,\theta +f=mg\sin \,\theta +\mu \text{ }mg\cos \,\theta \]
\[=mg\cos
\theta =mg\,(\sin \theta +\mu \,\cos \theta )\]
(d)
Force oil the box, when moves with an acceleration \[\alpha ,\,\,{{f}_{1}}=ma\]
\Force applied upward
\[=F+{{f}_{1}}=mg(\sin
\theta +\mu \cos \theta )+ma\]
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