Question

A book of mass 7 kg rests on a plank. You tilt one end of the plank and slowly increase the angle of the tilt. The coefficient of static friction between the book and the plank is 0.32. What is the maximum angle of tilt for which the book will remain stationary and not slide down the plank

Answer 1

Answer:

17.74°

Explanation:

For an inclined surface,

F = μmgcosΦ................ Equation 1

F' = mgsinΦ................. Equation 2

Where F = friction force, F' = down slope force, m = mass of the book, g = acceleration due to gravity, μ = coefficient of static friction, Φ = maximum angle of tilt.

For the book to remain stationary and not slide.

F = F'

μmgcosΦ = mgsinΦ

μ = cosΦ/sinΦ

μ = tanФ

make Ф the subject of the equation

Ф = tan⁻¹(μ).................... Equation 3

Given: μ = 0.32

Substitute into equation 3

Ф = tan⁻¹(0.32)

Ф = 17.74°

Answer 2

To solve this problem we will apply the concepts related to the Friction force and the force induced by gravity. Since the displacement is in angular mode, the component of the horizontal force of friction will be equivalent to the component of the vertical force of gravity. For balance to exist, both must be equal

$F_{fx} = F_{gy}$

Where,

$F_{fx} = \mu mg cos\theta$

$F_{gy} = mg sin \theta$

m = Mass

g = Gravitational acceleration

$\mu$ = Constant of friction

Then,

$m g sin \theta = \mu m g cos \theta$

$sin \theta = mu cos \theta$

$tan \theta = mu$

$tan\theta = 0.32$

$\theta = 17.74\°$

Therefore the maximum angle  of tilt for which the book will remain stationary and not slide down the plank is 17.74°