Question

If a pushing force an angle $\alpha$ with horizontal is applied on a block of mass $m$ placed on horizontal table and angle of function is $\beta$, then minimum magnitude of force required to move the block is

• A. $\frac{mg\sin \beta }{\cos \left[\alpha -\beta \right]}$
• B. $\frac{mg\sin \beta }{\cos \left[\alpha +\beta \right]}$
• C. $\frac{mg\sin \beta }{\sin \left[\alpha +\beta \right]}$
• D. $\frac{mg\cos \beta }{\cos \left[\alpha -\beta \right]}$

Answer 1

The correct option is D $\frac{mg\cos \beta }{\cos \left[\alpha -\beta \right]}$

$\mu =\tan \beta$
$N=mg+F\cos \alpha$
so, $F\sin \alpha =\mu N$
$F\sin \alpha =\mu \left[mg+F\cos \alpha \right]$
$F\left[\sin \alpha -\mu \cos \alpha \right]=\mu mg$
$F=\frac{\mu mg}{\sin \alpha -\mu \cos \alpha }$
$=d\frac{\tan \beta \left(mg\right)}{\sin \alpha -\tan \beta \cos \alpha }$
$=\frac{\sin \beta \left(mg\right)}{\sin \alpha \cos \beta -\sin \beta \cos \alpha }$
$=\frac{\sin \beta \left(mg\right)}{\sin [\alpha -\beta ]}$