Problem 29 - University Physics with Modern Physics

Question

A 0.60\(\cdot \mathrm{kg}\) book slides on a horizontal table. The kinetic friction force on the book has magnitude 1.2 \(\mathrm{N}\) . (a) How much work is done on the book by friction during a displacement of 3.0 \(\mathrm{m}\) to the left? (b) The book now slides 3.0 \(\mathrm{m}\) to the right, returning to its starting point. During this second 3.0 \(\mathrm{m}\) 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) -3.6 J; (b) -3.6 J; (c) -7.2 J; (d) Nonconservative force.

Step 1: Calculate Work Done (Leftward)

The work done by friction when an object is moved is given by the formula \( W = F_d \cdot d \cdot \cos(\theta) \), where \( F_d \) is the force of friction, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement. In this scenario, friction acts opposite to the direction of displacement, so \( \theta = 180^\circ \), making \( \cos(\theta) = -1 \). Thus, the work done by friction is \( W = 1.2 \ \text{N} \times 3.0 \ \text{m} \times (-1) = -3.6 \ \text{J} \).

Step 2: Calculate Work Done (Rightward)

The book now moves 3.0 m to the right, again in the opposite direction to friction. The conditions are the same as in the previous step, with the work done by friction being \( W = 1.2 \ \text{N} \times 3.0 \ \text{m} \times (-1) = -3.6 \ \text{J} \).

Step 3: Calculate Total Work Done for Round Trip

For the complete round trip (3.0 m to the left and 3.0 m to the right), add the work done during both displacements: \( W_{\text{total}} = -3.6 \ \text{J} + (-3.6 \ \text{J}) = -7.2 \ \text{J} \).

Step 4: Determine Nature of Friction Force

Friction is nonconservative. A conservative force does zero total work on a closed path; however, friction has done \(-7.2 \ \text{J}\) of work during the round trip, showing it is nonconservative.

Key Concepts

Kinetic Friction

Kinetic friction is a type of friction that occurs between moving surfaces. When you slide an object over a surface, kinetic friction is the force that resists the motion. This force acts in the opposite direction to the movement. It is proportional to the normal force, the force perpendicular to the contact surface, experienced by the object.

For our book sliding across the table, the force of kinetic friction is given as 1.2 N. This value does not change with the speed of the book and remains constant as long as the surfaces and conditions remain unchanged.

• Kinetic friction depends on both the surfaces in contact and the normal force.
• It causes mechanical energy to convert into thermal energy, thus opposing the sliding motion.

In our example, the kinetic friction causes a negative work of -3.6 J to occur as the book moves in either direction. Whenever movement happens, kinetic friction ensures some energy from the system is lost as heat.

Nonconservative Forces

Nonconservative forces, like friction, are forces where the work they do on a moving object cannot be recovered fully upon the return to an original position. In essence, the energy is lost from the system, often as heat or sound, in a manner that is not reversible.

• Unlike conservative forces, nonconservative forces depend on the path taken, not just start and end positions.
• They continuously extract energy from moving bodies, reducing mechanical energy overall.

In our exercise, the frictional force is categorized as nonconservative. This is because during the full round trip of the book sliding back and forth, a total of -7.2 J of work is done. The energy expended through work against friction isn't recovered when returning to the starting point, confirming the nonconservative nature of kinetic friction.

Work-Energy Principle

The work-energy principle is fundamental in physics and relates the work done on an object to its change in kinetic energy. According to this principle, if a force does work on an object, it changes the object's kinetic energy by the same amount.

• Work done by nonconservative forces leads to changes in both kinetic energy and thermal energy.
• In our scenario, all the work done by the frictional force converts mechanical energy into thermal energy, so the book's kinetic energy reduces.

When the book advances 3 meters to the left and the friction does -3.6 J of work, that energy is removed from the book's kinetic energy. The same holds true when moving to the right. The work-energy principle helps quantify these energy transformations and enforces a methodical approach to understanding the motion affected by forces like friction.