Problem 30 - University Physics with Modern Physics

Question

You and three friends stand at the corners of a square whose sides are 8.0 \(\mathrm{m}\) long in the middle of the gym floor, as shown in Fig. \(7.26 .\) 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 \(\mu_{\mathrm{x}}=0.25 .\) (a) The book slides from you to Beth and then from Beth to Carlos, along the lines connecting 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 friction 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) 41.54 J, (c) 58.8 J, (d) Non-conservative.

Step 1: Calculating the force of friction

The frictional force can be expressed using the formula: \[ f_k = \mu_k \cdot m \cdot g \]where \( f_k \) is the force of friction, \( \mu_k = 0.25 \) is the coefficient of kinetic friction, \( m = 1.5 \, \mathrm{kg} \) is the mass, and \( g = 9.8 \, \mathrm{m/s^2} \) is the acceleration due to gravity. Substituting these in, we get:\[ f_k = 0.25 \cdot 1.5 \cdot 9.8 = 3.675 \, \mathrm{N} \]

Step 2: Work done by friction (a) - Linear path

The work done by friction when sliding along the square's side is computed by:\[ W = f_k \cdot d \]where \( d = 8 \times 2 = 16 \, \mathrm{m} \) (the total length from you to Beth and then to Carlos, since they are adjacent sides). Thus, \[ W = 3.675 \cdot 16 = 58.8 \, \mathrm{J} \].

Step 3: Work done by friction (b) - Diagonal path

For the diagonal path, the diagonal length \( d \) in a square is given by:\[ d = s \cdot \sqrt{2} \]where \( s = 8 \, \mathrm{m} \). So, the diagonal is:\[ d = 8 \cdot \sqrt{2} = 11.31 \, \mathrm{m} \]. Thus, the work done is:\[ W = 3.675 \cdot 11.31 = 41.54 \, \mathrm{J} \].

Step 4: Work done by friction (c) - Round trip

If the book is slid from you to Kim and then back, the total path length \( d \) is \( 8 \cdot 2 = 16 \, \mathrm{m} \) as Kim is opposite to your initial position. The work done during this path is:\[ W = 3.675 \cdot 16 = 58.8 \, \mathrm{J} \].

Step 5: Nature of the frictional force

Frictional force is non-conservative because the work done by friction depends on the path taken and not just on the initial and final positions. Energy is dissipated as heat, and cannot be recovered.

Key Concepts

Kinetic friction

Kinetic friction is an important concept in physics that deals with the resistive force that acts between surfaces in relative motion. Whenever two surfaces slide against each other, kinetic friction comes into play. This friction is responsible for slowing down the motion and eventually bringing it to a halt if no other external forces are applied.

• The coefficient of kinetic friction, denoted as \( \mu_k \), quantifies how much resistance one surface offers against another. It varies depending on the material properties of the surfaces in contact.
• In the problem at hand, the coefficient is given as 0.25, which tells us that the friction between the book and the gym floor is moderate.
• The force of kinetic friction \( f_k \) can be calculated using the formula: \[ f_k = \mu_k \times m \times g \] where\( m \) is the mass of the object and \( g \) is the acceleration due to gravity.

Here's how it affects movement: if you push a book across a table, kinetic friction is the force that resists its motion. Understanding this force helps us comprehend real-world situations like driving or walking, where friction plays a crucial role.

Conservative vs non-conservative forces

In physics, understanding the distinction between conservative and non-conservative forces is crucial. Conservative forces, like gravitational force, depend only on the initial and final positions of an object. This means that the work done by conservative forces is reversible, and the energy is conserved, often being converted between potential and kinetic forms.

Non-conservative forces, however, like friction, depend on the path taken by the object. This characteristic implies that energy is lost from the system, usually as heat or sound, and cannot be fully recovered. Friction is a classic example of a non-conservative force since the work done by friction is different depending on the path and cannot be undone.

• Friction's work is always negative because it acts opposite to the direction of motion, removing energy from the moving object.
• Unlike conservative forces, no potential energy is stored due to friction. It converts the object’s kinetic energy directly into heat.
• In the exercise provided, since the work done by friction changes based on the path traveled, it confirms the non-conservative nature of this force.

Physics problem solving

When tackling physics problems, a systematic approach can significantly ease the process and enhance comprehension. Here's a simple strategy to follow:

• Identify the Problem: Determine what is being asked. Establish known values (like mass, coefficient of friction, distance) and what needs to be found (e.g., work done by friction).
• Visualize the Scenario: Drawing a diagram can help conceptualize the problem and lead to better problem-solving strategies.
• Apply Relevant Equations: Use the formulas applicable to the problem at hand. For friction problems, the main formula is \( f_k = \mu_k \times m \times g \) and work done is \( W = f_k \times d \).
• Calculate Step-by-step: Substitute the known values into the equations and solve systematically. Breaking down complex problems into smaller, manageable parts can prevent mistakes and confusion.
• Review and Interpret: After solving, examine the results for reasonableness and ensure they answer the original question completely.

By following these steps, students can approach physics problems more effectively, improving their understanding and ability to find the correct solutions.

Force of friction calculation

Calculating the force of friction is essential in solving many physics problems involving motion. Here are the steps to calculate the force of kinetic friction, such as in the problem given:

• Determine the mass \( m \) of the object. For our case, the book has a mass of 1.5 kg.
• Identify the coefficient of kinetic friction \( \mu_k \) between the contact surfaces, in this instance, \( \mu_k = 0.25 \).
• Recognize that the normal force \( N \), which is the force perpendicular to the contact surface, is equal to \( m \times g \) for horizontal surfaces. Here, it equals \( 1.5 \times 9.8 = 14.7 \; \mathrm{N} \).
• Apply the formula: \( f_k = \mu_k \times N \), giving \( f_k = 0.25 \times 14.7 = 3.675 \; \mathrm{N} \).

Once the force of friction is found, calculating the work done involves multiplying this force by the distance over which it acts. The simplicity of this calculation hides the complexity of real-world applications, where understanding friction can be critical, whether it’s for designing machinery or understanding natural phenomena.