Question

A particle of mass 2kg is placed on a rough plane which is inclined at 30°to the horizontal. A force of 20N acts on the particle in a direction parallel to and up the plane. If the particle is just about to move up the plane, analytically obtain the coefficient of friction. (g=10m/s^2)

Answer 1

According to Newton's second law, the particle is in rest (which we can compare to "about to move") if all forces are in equilibrium:

$$Ox: -mg\text{ sin}\theta-f+F=0,\\ Oy: N-mg\text{ cos}\theta=0,\\ f=\mu N.$$

Therefore, we see that the force force of friction can be expressed as

$$f=\mu mg\text{ cos}\theta,$$

the coefficient of friction can be expressed as

$$\mu=\frac{f}{mg\text{ cos}\theta}=\frac{F-mg\text{ sin}\theta}{mg\text{ cos}\theta}.\\\space\\ \mu=\frac{20-2\cdot10\text{ sin}30°}{2\cdot10\text{ cos}\theta}=0.58.$$