## Minimum Force Needed To Move Block On Rough Horizontal Surface

What is the minimum force required to move a block mass
$m$
along a rough horizontal surface with coefficient or friction
$\mu$
?
Let
$F$
be a force at
$\theta$
to the horizontal and let
$R$
be the normal reaction force.

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