Question

The smallest angle of inclined desk at which a book begins to slide off the desk is 15∘ relative to the horizontal.

Answer 1

The question appears to be incomplete.
I assume that we are to find the coefficient of static friction, μ, between the desk and the book.

Refer to the diagram shown below.
m = the mass of the book
mg = the weight of the book (g = acceleration due to gravity)

N = the normal reaction, which is equal to
N = mg cos(12°)

R =  the frictional force that opposes the sliding down of the book. It is
R = μN = μmg cos(12°)

F =  the component of the weight acting down the incline. It is
F = mg sin(12°)

Because the book is in static equilibrium (by not sliding down the plane), therefore
F = R
mg sin(12°) = μmg cos(12°)
$\frac{sin(12^{o})}{cos(12^{o})} = \frac{\mu mg}{mg} = \mu \\\\ tan(12^{o}) = \mu$

Therefore, the static coefficient of friction is
μ = tan(12) = 0.213

Answer:  μ = 0.21 (nearest tenth)

Answer 2

The coefficient of friction between the book and the desk is $\boxed{0.268}$ .

Further Explanation:

When a body starts moving on an inclined plane at the minimum inclination, then the inclination of the plane at that instant is termed as the angle of repose. At the angle of repose, the sliding of the body kept on the inclined plane starts.

At this instant, the expression of force balancing on the book can be written as:

$\boxed{{F_f}=mg\sin \theta }$

Here, ${F_f}$  is the frictional force experienced by the book and $mg\sin \theta$  is the weight of the book acting in the downward direction.

The friction force acting on the book in the upward direction along the inclined plane is:

$\begin{aligned}{F_f}&=\mu N\\&=\mu mg\cos \theta \\\end{aligned}$

Substitute the value of the friction force acting on the book.

$\begin{aligned}\mu mg\cos\theta=mg\sin \theta\\\mu=\tan \theta \\\end{aligned}$

Substitute the value of $\theta$  in equation (1).

$\begin{aligned}\mu &=\tan \left({15}\right)\\&=0.267\\\end{aligned}$

Thus, the coefficient of friction between the book and the desk is $\boxed{0.268}$ .

Learn More:

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Answer Details:

Grade: Middle School

Subject: Physics

Chapter: Laws of Motion

Keywords:

Smallest angle, inclined desk, angle of repose, friction force, begins to slide, off the desk, inclined plane, relative to horizontal, coefficient of friction.