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

Proof:

Let the mass of the box be mm

The diagram below will be very helpful:

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

As net force is 00, therefore mgsin(p)=µmgcos(p)\cancel{mg}\sin(p)=µ\cancel{mg}\cos(p) sin(p)=µcos(p)\sin(p)=µ\cos(p) µ=tan(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!

Note by Zakir Husain
4 months ago

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