Question

A block is moving on an inclined plane making an angle ${45}^0$ with the horizontal and the coefficient of friction is $\mu$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down.If we define $N=10\mu$, then N is

Answer 1

$\left(i\right)$ when force is applied up along the slope to push if up

As $N=mg$

$f_1=\mu +\frac{mg}{12}=\mu \frac{mg}{\sqrt{2}}+\frac{mg}{\sqrt{2}}$

$\left(ii\right)$ when the force is applied up the slope to prevent sliding

Here also,

$N=mg$

$-f_{2}mg=\mu N$

$\Rightarrow f_2=-\mu \frac{mg}{\sqrt{2}}+\frac{mg}{\sqrt{2}}$

It is given that.

$f_1=3f_2$

So, $\mu g+\frac{mg}{\sqrt{2}}=3\frac{mg}{\sqrt{2}}-3\mu \frac{mg}{\sqrt{2}}$

$\Rightarrow \mu +1=3-3\mu$

$\Rightarrow \mu =2/4=0.5$

So, $O\mu$ is $5$