Problem 26 - College Physics

Question

A \(2 \mathrm{~kg}\) book sits at rest on a horizontal table. The coefficient of static friction between the book and the surface is \(0.40,\) and the coefficient of kinetic friction is 0.20 . (a) What is the normal force acting on the book? (b) Is there a friction force on the book? (c) What minimum horizontal force would be required to cause the book to slide on the table? (d) If you give the book a strong horizontal push so that it begins sliding, what kind of force will cause it to come to rest? (e) What is the magnitude of this force?

Step-by-Step Solution

Answer

(a) 19.6 N; (b) No, it is 0 N; (c) 7.84 N; (d) Kinetic friction; (e) 3.92 N.

Step 1: Calculating the Normal Force

For an object resting on a horizontal surface, the normal force \( N \) is equal to the gravitational force acting on the object. The gravitational force \( F_g \) is calculated using the equation \( F_g = m \cdot g \), where \( m = 2 \ kg \) is the mass of the book and \( g = 9.8 \ m/s^2 \) is the acceleration due to gravity. Thus, \( F_g = 2 \ kg \cdot 9.8 \ m/s^2 = 19.6 \ N \). Therefore, the normal force \( N \) is \( 19.6 \ N \).

Step 2: Checking for Static Friction Force

The static frictional force \( f_s \) prevents an object from moving and acts only if an external force is applied. Since the book is at rest, and no horizontal force is currently applied, the static frictional force is \( 0 \ N \) because no external force is acting to overcome by friction yet.

Step 3: Finding the Minimum Force to Overcome Static Friction

The maximum static frictional force \( f_{s, max} \) can be calculated using \( f_{s, max} = \mu_s \cdot N \), where \( \mu_s = 0.40 \) is the coefficient of static friction. Substituting the known values: \( f_{s, max} = 0.40 \cdot 19.6 \ N = 7.84 \ N \). Therefore, a minimum horizontal force of \( 7.84 \ N \) is required to start moving the book.

Step 4: Understanding the Force to Stop the Book

Once the book starts sliding, the kinetic frictional force \( f_k \) will oppose the motion and eventually bring the book to rest. This happens because the kinetic friction continuously opposes the motion of the sliding book.

Step 5: Calculating the Kinetic Frictional Force

The kinetic frictional force \( f_k \) is calculated using \( f_k = \mu_k \cdot N \), where \( \mu_k = 0.20 \) is the coefficient of kinetic friction. Substituting the known values: \( f_k = 0.20 \cdot 19.6 \ N = 3.92 \ N \). This is the force that acts to slow down and eventually stop the sliding book.

Key Concepts

Static Friction

Static friction is the force that prevents an object at rest from starting to move when a horizontal force is applied. It is like an invisible glue that holds an object in place. This force only acts when there is a tendency for relative motion between two surfaces that are in contact. It increases to a maximum value based on the normal force and the coefficient of static friction.
This maximum limit is calculated as: \( f_{s, max} = \mu_s \cdot N \). Here, \( \mu_s \) is the coefficient of static friction, and \( N \) is the normal force. For the book in the exercise, static friction will initially be zero because no force is applied yet. However, if you start pushing the book, static friction will increase until it is overcome by reaching its maximum, beyond which the book starts moving.

Kinetic Friction

Once the book starts moving, static friction no longer applies, and kinetic friction comes into play. This frictional force acts when an object is sliding over a surface, continuously opposing the motion, which differs from static friction because it remains constant regardless of the velocity.
Kinetic friction is often weaker than static friction, which is why it is easier to keep something sliding once it has started. The kinetic frictional force can be calculated using: \( f_k = \mu_k \cdot N \), where \( \mu_k \) is the coefficient of kinetic friction. For exercise purposes, this ensures that the book slows down and eventually stops.

Normal Force

The normal force is crucial in understanding friction as it represents the perpendicular force exerted by a surface upon an object resting on it. Think of it as the force a table provides to support a book resting on it. The normal force is essential in calculating both static and kinetic friction because it determines the maximum frictional force that can occur before motion starts.
In scenarios where the surface is horizontal and the object is at rest, the normal force is equal to the object's weight. For the book in the exercise, this force was calculated using the mass of the book and gravitational acceleration: \( N = m \cdot g = 19.6 \ N \). This force serves as a baseline to compute the frictional forces.

Horizontal Force

A horizontal force is any force applied parallel to the surface on which an object rests. It is significant in initiating and sustaining the motion. In the context of the exercise, a minimum horizontal force must overcome static friction to start moving the object.

• To begin moving the book, a horizontal force greater than the maximum static friction force \( f_{s, max} \) needs to be applied.
• Once the book moves, maintaining or increasing horizontal force is necessary to counter kinetic friction and continue the book's motion.

This demonstrates the role horizontal forces play in both starting and managing motion dynamics.

Motion Dynamics

Motion dynamics refers to the interplay of forces that cause and manage motion. It's about how and why objects move and stop moving. Understanding friction is crucial to motion dynamics, as it helps dictate when an object begins to move and how it will stop.

• The stepping from static friction to kinetic friction is a fundamental part of this dynamics.
• Motion dynamics also includes calculating forces needed to alter states of rest or motion.

In the problem, understanding these dynamics was key to calculating the forces required for the book's motion and eventual stop.