Problem 27 - University Physics with Modern Physics

Question

A 0.60-kg book slides on a horizontal table. The kinetic friction force on the book has magnitude 1.8 N. (a) How much work is done on the book by friction during a displacement of 3.0 m to the left? (b) The book now slides 3.0 m to the right, returning to its starting point. During this second 3.0-m displacement, how much work is done on the book by friction? (c) What is the total work done on the book by friction during the complete round trip? (d) On the basis of your answer to part (c), would you say that the friction force is conservative or nonconservative? Explain.

Step-by-Step Solution

Answer

(a) -5.4 J, (b) -5.4 J, (c) -10.8 J, (d) Nonconservative force.

Step 1: Identify Friction Work Formula

Work done by a constant force is given by the formula:\[ W = F \cdot d \cdot \cos(\theta) \]where \( W \) is the work done, \( F \) is the magnitude of the force, \( d \) is the displacement, and \( \theta \) is the angle between the force and the direction of displacement. In this case, since kinetic friction is acting opposite to the direction of displacement, \( \theta = 180^\circ \). Hence, \( \cos(180^\circ) = -1 \).

Step 2: Calculate Work Done Moving Left

Initially, the book moves 3.0 m to the left against a friction force of 1.8 N:\[ W = 1.8 \times 3.0 \times \cos(180^\circ) = 1.8 \times 3.0 \times (-1) = -5.4 \text{ J} \]Thus, the work done by friction on the book moving to the left is \(-5.4\) Joules.

Step 3: Calculate Work Done Moving Right

When the book moves 3.0 m back to the right, the friction force still acts against the motion:\[ W = 1.8 \times 3.0 \times \cos(180^\circ) = 1.8 \times 3.0 \times (-1) = -5.4 \text{ J} \]The work done by friction on the book moving to the right is \(-5.4\) Joules.

Step 4: Calculate Total Work for Round Trip

Add the work done for both 3.0 m segments (to the left and back to the right):\[ W_{\text{total}} = -5.4 + (-5.4) = -10.8 \text{ J} \]The total work done by friction during the complete round trip is \(-10.8\) Joules.

Step 5: Determine Nature of Friction Force

The total work done by friction over the round trip is not zero \((-10.8 \text{ J})\). This indicates that energy is not conserved in the presence of friction—friction converts mechanical energy into thermal energy. Thus, the friction force is a nonconservative force.

Key Concepts

Nonconservative Forces

In physics, forces are divided into two categories: conservative and nonconservative forces. When dealing with nonconservative forces, one key aspect is that they cause a change in mechanical energy which cannot be recovered by simply reversing the path. Friction is a prime example of a nonconservative force.
Though a conservative force does not depend on the path taken but only on the initial and final positions, nonconservative forces like friction depend greatly on the path traveled. This means that the work done by friction can vary depending on the length and direction of the path.
Therefore, in problems involving friction as we have seen in this exercise, the total mechanical energy of a system does not remain constant. Instead, energy is dissipated, often as heat, due to the frictional force acting on moving objects.

Work Done by Friction

Work is the energy transferred to or from an object via a force acting upon it. In terms of friction, work done is calculated with the formula:

• \( W = F \cdot d \cdot \cos(\theta) \)

Here, \( F \) represents the frictional force, \( d \) is the displacement, and \( \theta \) is the angle between force direction and displacement. In our case with kinetic friction, \( \theta \) is \( 180^\circ \) because the force is opposite to the direction of movement.
From the calculation in the problem, we see that the work done by friction when moving to the left is \(-5.4 \text{ J}\), while moving back to the right is also \(-5.4 \text{ J}\). This results in a total work of \(-10.8 \text{ J}\) for the round trip. Negative work indicates that friction is taking energy out of the system, opposing the motion at all times and converting energy into other forms like heat.

Mechanical Energy Conversion

Mechanical energy in a system consists of both kinetic and potential energy. When nonconservative forces such as friction act on a system, they convert mechanical energy into other forms of energy, typically thermal energy.
In the exercise, the negative work done by friction decreases the total mechanical energy of the system. This conversion is irretrievable, meaning that once mechanical energy is turned into heat by friction, it cannot be turned back again within the same system.
This energy transformation underscores why understanding nonconservative forces is vital. They remove energy from mechanical stores, preventing energy conservation and affecting calculations in dynamic systems or engineering contexts. Hence, friction plays a critical role not just in physical interactions but also in energy management and inefficiencies.