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 0.46. What is the maximum angle of tilt for which the book will remain stationary and not slide down the plank?

To find the maximum angle of tilt for which the book will remain stationary and not slide down the plank, we can use the concept of static friction and the conditions for equilibrium.

The static friction force acts parallel to the surface between the book and the plank and opposes the tendency of the book to slide. It can be calculated using the formula:

F_friction = coefficient of static friction * normal force

In this case, the normal force acting on the book is equal to its weight.

normal force = mass of the book * acceleration due to gravity

normal force = 7 kg * 9.8 m/s^2

Plugging the values, we get:

normal force = 68.6 N

Next, we need to find the maximum angle of tilt. To do that, we need to consider the forces acting on the book when it is at the verge of sliding down.

The gravitational force acting on the book can be divided into two components: one perpendicular to the plank and the other parallel to the plank. The component parallel to the plank is given by:

F_parallel = mass of the book * acceleration due to gravity * sin(angle)

We set the parallel component equal to the maximum static friction force to find the maximum angle:

F_parallel = F_friction

mass of the book * acceleration due to gravity * sin(angle) = coefficient of static friction * normal force

7 kg * 9.8 m/s^2 * sin(angle) = 0.46 * 68.6 N

Simplifying the equation:

sin(angle) = (0.46 * 68.6 N) / (7 kg * 9.8 m/s^2)

sin(angle) ≈ 0.211

To find the angle, we take the inverse sine (sin^-1) of both sides of the equation:

angle ≈ sin^-1(0.211)

Using a calculator, we find that:

angle ≈ 12.3 degrees

Therefore, the maximum angle of tilt for which the book will remain stationary and not slide down the plank is approximately 12.3 degrees.