Problem 58 - College Physics

Question

\(\bullet\) \(\bullet\) You and three friends stand at the corners of a square whose sides are 8.0 \(\mathrm{m}\) long in the mid- dle of the gym floor, as shown in the accompanying figure. You take your physics book and You take your physics book and push it from one person to the other. The book has a mass of \(1.5 \mathrm{kg},\) and the coefficient of kinetic friction between the book and the floor is \(\mu_{k}=0.25\) . (a) The book slides from you to Beth and then from Beth to Carlos, along the lines connect- ing these people. What is the work done by friction during this displacement? (b) You slide the book from you to Carlos along the diagonal of the square. What is the work done by friction during this displacement? (c) You slide the book to Kim who then slides it back to you. What is the total work done by fric- tion during this motion of the book? (d) Is the friction force on the book conservative or nonconservative? Explain.

Step-by-Step Solution

Answer

(a) 58.8 J, (b) 41.56 J, (c) 58.8 J, (d) Non-conservative force.

Step 1: Understand the problem layout

Imagine the square as a grid with you and your friends standing at each corner. Let's label the corners as A, B, C, and D. Your movements will involve sliding the book along the sides and diagonals of the square.

Step 2: Calculate the work done by friction (side length paths)

For parts (a), consider moving the book from you (A) to Beth (B), and then from Beth (B) to Carlos (C). Total path length = 8.0 m + 8.0 m = 16.0 m. Use the formula for work done by friction: \[ W = f_k \cdot d \] where \( f_k = \mu_k \cdot m \cdot g \), with \( \mu_k = 0.25 \), \( m = 1.5 \ \text{kg} \), and \( g = 9.8 \ \text{m/s}^2 \).Thus, \( f_k = 0.25 \times 1.5 \times 9.8 = 3.675 \ \text{N} \).Desplacement \( d = 16.0 \ \text{m} \). Therefore,\[ W = 3.675 \times 16.0 = 58.8 \ \text{J} \]

Step 3: Calculate work done by friction (diagonal path)

For part (b), determine the diagonal of the square using Pythagoras Theorem: \[ \text{Diagonal} = \sqrt{8.0^2 + 8.0^2} = 11.31 \text{ m} \]Then compute the work done using the diagonal path:\[ W = f_k \cdot d = 3.675 \times 11.31 \approx 41.56 \ \text{J} \]

Step 4: Calculate total work from back-and-forth motion

For part (c), move the book from corner A to Kim (D) and then back to A. Each side is 8.0 m, so the round trip is 16.0 m. Work done is calculated again as:\[ W = f_k \cdot d = 3.675 \times 16.0 = 58.8 \ \text{J} \]

Step 5: Determine the nature of the friction force

For part (d), understand that the work done by friction depends on the path taken and is not recoverable. Thus, friction is a non-conservative force.

Key Concepts

Work Done by Friction

When a book slides across a surface, work is done against the friction between the book and the floor. **Friction** is a force that resists the motion of two surfaces sliding past each other. The **work done by friction** is calculated using the formula:
\[ W = f_k \cdot d \]where:

• \( W \) is the work done by friction, expressed in Joules (J)
• \( f_k \) is the force of kinetic friction, i.e., the friction acting when the book is in motion.
• \( d \) is the displacement, or the distance over which the book slides.

The kinetic friction force \( f_k \) can be calculated by the equation:
\[ f_k = \mu_k \cdot m \cdot g \]where:

• \( \mu_k \) is the coefficient of kinetic friction (a unitless number that represents the friction between two surfaces).
• \( m \) is the mass of the book in kilograms.
• \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \).

The work done by friction is always negative because the friction force opposes the direction of motion.

Kinetic Friction

**Kinetic friction** is the frictional force acting between moving surfaces. It plays a crucial role by opposing the motion of an object sliding over another. Unlike static friction, which does not come into play until an object begins to move, kinetic friction is active as long as there is relative motion.
The formula to calculate kinetic friction is:
\[ f_k = \mu_k \cdot N \]where \( N \) is the normal force, which is usually equal to the product of the mass \( m \) of the object and the acceleration due to gravity \( g \), thus:
\[ N = m \cdot g \]For example, when a person pushes a book across the gym floor, the force that counters this movement is kinetic friction. The amount of kinetic friction depends on the nature of the surfaces in contact and is characterized by the **coefficient of kinetic friction** \( \mu_k \). This coefficient varies between different materials, indicating how difficult it is to slide one surface over another. A higher \( \mu_k \) means more resistance.

Non-Conservative Forces

In physics, forces can be either conservative or non-conservative. A **non-conservative force** is one for which the work done depends on the path taken. Unlike conservative forces, which are path-independent, non-conservative forces such as friction cause energy to be dissipated, usually as heat.
For example, if you slide a book along a path, the work done by friction does not depend solely on the start and end points, but on the distance the book travels and the path it takes.

• If you slide a book direct along a diagonal path, less work is done by friction compared to moving it around the square sides, even if both start and end at the same points.
• This is because, with non-conservative forces, energy is not stored or recoverable; it is lost to the environment, making the friction force non-recoverable.

Therefore, when non-conservative forces like friction are involved, energy conservation equations cannot be applied directly as they are with conservative forces.# Understanding such forces is key in solving real-world problems, as they account for the disparity between theoretical predictions and actual outcomes due to energy dissipation.