It's slope!

After working for sometime on friction I got the following result:

Suppose there is a box on top of a tilted surface and coefficient of static friction between them is $µ$ . Now suppose the angle at which the surface is inclined to ground is $\theta$ and you are increasing it slowly, after reaching a certain value $\theta=p$ the force of friction reaches it's maximum point (i.e. if you now increase $\theta$ a little more, the box will fall). Then the following relation holds true: $µ=\tan(p)$

Proof:

Let the mass of the box be $m$

The diagram below will be very helpful:

The force of friction acting $=µN=µmg\cos(p)$

As net force is $0$ , therefore $\cancel{mg}\sin(p)=µ\cancel{mg}\cos(p)$ $\sin(p)=µ\cos(p)$ $\boxed{µ=\tan(p)}$

Note:

I haven't proved the diagram because I found it very easier to proof, if you want you can proof it yourself and write it in the comment it can be challenge for you!

#Mechanics

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$