A 310-g paperback book rests on a 1.2 -kg textbook. A force...

A $310-\mathrm{g}$ paperback book rests on a 1.2 -kg textbook. A force is applied to the textbook, and the two books accelerate together from rest to $96 \mathrm{cm} / \mathrm{s}$ in 0.42 s. The textbook is then brought to a stop in $0.33 \mathrm{s}$ during which time the paperback slides off. Within what range does the coefficient of static friction between the two books lie?

Key Concepts

Free-Body Diagram and Force Analysis

Drawing a free-body diagram is a fundamental technique in mechanics used to visualize the forces acting on an object. By representing all forces, such as applied forces, frictional forces, gravitational forces, and normal forces, one can set up equations (often derived from Newton's laws) to analyze the motion and predict outcomes like sliding or staying in place. This method is crucial for understanding how different forces interact in the system.

Kinematics (Acceleration and Deceleration)

Kinematics involves the study of motion without considering the causes of that motion. In problems where an object accelerates or decelerates over a period of time, calculating the acceleration by dividing the change in velocity by the time interval is essential. This acceleration can then be used in dynamics analyses to relate forces to changes in velocity.

Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the product of its mass and acceleration. This law is central to solving problems involving forces and motion, as it provides the relationship used to calculate the acceleration of an object when a force is applied, or conversely, to determine the necessary force to achieve a desired acceleration.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. Its maximum value is proportional to the normal force exerted between the surfaces and is characterized by the coefficient of static friction. Determining this coefficient allows one to assess whether the frictional force is sufficient to prevent relative motion under an applied force.

Transcript

00:01 So we have a situation where we have two books and we are accelerating them to a certain speed in a certain amount of time.

00:11 And we want them to be able to accelerate together.

00:15 And when we brought to a stop, the top one slides off.

00:20 So we want to know what is the static friction coefficient between the books.

00:24 So this feature can happen.

00:26 So we have two situations.

00:27 The first one, we want the.

00:29 The static friction to be able to hold the books together when we are accelerating to the right.

00:36 So we have gravity and normal force.

00:41 We have the acceleration to the right and the static friction opposing the direction that the other book is trying to go, that is to the left.

00:53 So fs1.

00:57 This is the situation one.

00:59 So this is when both books are moving together.

01:02 And the situation too is when we stop the book from below, and then we want to be able to slide the book from above using the less friction possible.

01:20 So we have gravity and normal force, but now the static friction is pointing in the other direction because the acceleration is also pointing the other direction and the tendency of movement is to keep into the right.

01:34 Friction is always opposite to the direction of the velocity, and in this case, the velocity is pointed to the right, even though the acceleration is pointing to the left.

01:45 Everything here is vectors.

01:48 Now, the first point to solve this problem is to find what is the values of the acceleration, and this is easy to do because we have the information of the time and the velocity and the initial velocity of both cases.

02:03 So for case one, we have acceleration is v over t.

02:11 The models of the acceleration is v over t.

02:14 And then this is 0 .96 divided by 0 .42.

02:20 And this gives us 2 .286 meters per second square.

02:27 And for the this acceleration is the same thing.

02:31 So a2 is v0 over t...

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