Question

A block of mass m is placed on a rough horizontal surface. The coefficient of friction between them is $\mu$. An external horizontal force is applied to the block and its magnitude is gradually increased. The force exerted by the block on the surface is R then :

• A. the magnitude of R will gradually increase.
• B. $R\le mg\sqrt{{\mu }^2+1}$
• C. the angle made by R with the vertical will gradually increase.
• D. the angle made by R with the vertical $\le ta{n}^{-1}\mu$.

Answer 1

Answer: (A), (B), (D)

The correct options are
A the magnitude of R will gradually increase.
B $R\le mg\sqrt{{\mu }^2+1}$
D the angle made by R with the vertical $\le ta{n}^{-1}\mu$.

$N=Mg$;

$f_s=N\mu$

$f_s=mg\mu \to$ maximum value of friction possible

Now $R=$ resultant of $N\&f_s$

Now, value of $f_s$ keeps on increasing and it is equal to the external force $f$ until

$f=f_s$

after that friction force is constantly $mg\mu$

So $R$ increases gradually

$R=\sqrt{(mg\mu {)}^2+(mg{)}^2}$

$R=mg\sqrt{{\mu }^2+1}\to$ maximum value of $R$

Now $\tan \theta =\frac{f_s}{N}\Rightarrow \tan \theta =\frac{mgcl}{mg}$

$\Rightarrow \tan \theta =\mu$

$\theta$ (maximum value of $\theta$) $={\tan }^{-1}\left(\mu \right)$