Minimum Force Needed To Move Block On Rough Horizontal Surface

What is the minimum force required to move a block mass  
  along a rough horizontal surface with coefficient or friction  
  be a force at  
  to the horizontal and let  
  be the normal reaction force.

minimum force to move block on rough horizontal surface

Resolving horizontally and vertically:
\[\mu R =F cos \theta\]

\[R +F sin \theta =mg \rightarrow R=mg-F sin \theta\]

The first divided by the second gives  
\[\mu = \frac{F cos \theta }{mg- F sin \theta}\]
Solving for  
\[F= \frac{ \mu mg}{\mu sin \theta + cos \theta }\]
To find the minimum value of  
  note that the numerator is fixed, so we must maximise the denominator. This is maximum when  
\[\frac{d}{d \theta } (\mu sin \theta + cos \theta)= \mu cos \theta - sin \theta = 0\]
\[sin \theta = \mu cos \theta \rightarrow tan \theta = \mu\]
The minimum force is  
\[F= \frac{\mu mg}{cos \theta ( 1+ \mu tan \theta )}= \frac{ \mu mg sec \theta}{ 1+ \mu^2} = \frac{ \mu mg \sqrt{1+ tan^2 \theta}}{1+ \mu^2}= \frac{ \mu mg \sqrt{1+ \mu^2}}{1+ \mu^2}= \frac{ \mu mg}{\sqrt{1+ \mu^2}}\]

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