Question

A wooden block is kept on a polished wooden plank and the inclination of the plank is gradually increased. It is found that the block starts slipping when the plank makes an angle of ${18}^{\circ }$ (angle of repose) with the horizontal. However, once the block starts slipping it can continue with uniform speed if the inclination is reduced to ${15}^{\circ }$. Find the coefficients of static and kinetic friction between the block and the plank.

• A. = tan ⁼ tan
• B. = tan ⁼ tan
• C. = tan ( + ) ⁼ tan ( + )
• D. =

Answer 1

The correct option is A

= tan

⁼ tan

Free body drawing

N=mg cosθ

Since net external force in downward direction along the plane so tendency to slip is in that direction, therefore friction acts in opposite direction.

Since the block is about to slip so fr will be maximum i.e., limiting friction i.e., μ, N

N = mg cos $\theta$

mg sin $\theta$ = ${\mu }_s$ N

sin $\theta$ = ${\mu }_s$ cos$\theta$

${\mu }_s$ = tan$\theta$

${\mu }_s$ = tan${18}^{\circ }$

Now once the block is in motion kinetic friction comes into play which is little less than static so to just stop the acceleration of the block, the angle had to be decreased to ${15}^{\circ }$

N = mg cos $\theta$

${\mu }_K$ N = mg sin $\theta$

${\mu }_K$ = tan θ = tan${15}^{\circ }$