The 3.42 kg physics book shown is connected by a string to a 518.0 g coffee cup. The book is given a push up the slope and released with a speed of 2.73 m/s. The coefficients of friction are μs = 0.490 and μk = 0.185. What is the acceleration of the book if the slope is inclined at 27.4°?

5m/s^2

To find the acceleration of the book, we'll need to consider the forces acting on it. The forces in this case are the gravitational force, the normal force, and the frictional force.

First, let's calculate the force of gravity acting on the book. The force of gravity can be found using the formula:

F_gravity = mass * g

where mass is the mass of the book, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

F_gravity = 3.42 kg * 9.8 m/s^2 = 33.516 N

Next, we need to find the normal force. The normal force is the force exerted by a surface perpendicular to the object. In this case, the normal force is equal to the component of the gravitational force acting perpendicular to the slope. It can be calculated using the formula:

F_normal = F_gravity * cos(theta)

where theta is the angle of inclination of the slope.

F_normal = 33.516 N * cos(27.4°) = 29.439 N

Now, let's calculate the maximum static friction force. The maximum static friction force can be found using the equation:

F_static_friction = μs * F_normal

where μs represents the coefficient of static friction.

F_static_friction = 0.490 * 29.439 N = 14.427 N

Since the book is given a push up the slope and released, the frictional force acting on the book is kinetic friction. The kinetic friction force can be found using the equation:

F_kinetic_friction = μk * F_normal

where μk represents the coefficient of kinetic friction.

F_kinetic_friction = 0.185 * 29.439 N = 5.443 N

Now, we can calculate the net force acting on the book parallel to the slope. It is given by:

F_net = F_applied - F_kinetic_friction

Since the book is released without any additional applied force, the equation becomes:

F_net = - F_kinetic_friction (negative sign indicates opposing direction)

F_net = -5.443 N

Lastly, we can determine the acceleration of the book using Newton's second law of motion:

F_net = mass * acceleration

Rearranging the equation to solve for acceleration:

acceleration = F_net / mass

acceleration = -5.443 N / 3.42 kg

acceleration ≈ - 1.59 m/s^2

The negative sign indicates that the book is slowing down due to the opposing frictional force.

In which direction is the 2.73m/s velocity? Where is the coffee cup?