How to Find Energy Lost Due to Friction: A Comprehensive Guide

When it comes to understanding the principles of energy and its conservation, the concept of energy lost due to friction is a crucial aspect that every physics student must grasp. Friction, a ubiquitous force that opposes the relative motion between two surfaces, can lead to significant energy dissipation, primarily in the form of heat. In this comprehensive guide, we will delve into the intricacies of calculating the energy lost due to friction, providing you with a step-by-step approach and a wealth of technical details to enhance your understanding.

Calculating Energy Loss Due to Friction: The Fundamental Approach

The energy lost due to friction can be calculated using the formula for work done, which is the product of the force and the distance over which the force is applied. In the case of friction, the force is the frictional force, which can be calculated using the formula:

Frictional Force = Friction Coefficient × Normal Force

Once the frictional force is determined, the energy loss due to friction can be calculated by multiplying the frictional force by the distance over which the force is applied.

Energy Loss Due to Friction = Frictional Force × Distance

This fundamental approach forms the basis for understanding and calculating the energy lost due to friction in various scenarios.

Calculating Energy Loss in a Swing System

Let’s consider a practical example to illustrate the application of this principle. Imagine a 250 kg child on a 200 m long swing who is released. To calculate the mechanical energy lost due to friction, we need to follow these steps:

1. Determine the initial kinetic energy of the child:
   Kinetic Energy = 1/2 × Mass × Velocity^2
   where the velocity is the initial velocity of the child on the swing.

2. Determine the final kinetic energy of the child:
   Kinetic Energy = 1/2 × Mass × Velocity^2
   where the velocity is the final velocity of the child on the swing after it has come to a stop due to friction.

3. Calculate the energy lost due to friction:
   Energy Lost Due to Friction = Initial Kinetic Energy – Final Kinetic Energy

By following this approach, you can determine the mechanical energy lost due to friction in the swing system.

Calculating Viscous Energy Loss in the Left Ventricle (LV) of the Heart

Another example of energy loss due to friction can be found in the blood flow within the left ventricle (LV) of the heart. The blood flow in the LV is non-idealized, and with blood being a viscous fluid, frictional forces exist, resulting in irreversible mechanical energy loss mainly in the form of thermal energy (heat).

To calculate the viscous energy loss in the LV, we can use the viscous dissipation function Φ in the Newtonian Navier-Stokes energy equations. The viscous dissipation function Φ represents the rate of viscous energy dissipation per unit volume.

The total viscous energy loss in joules over a given period of time can be computed as:

Energy Loss (EL) = Σ (Φ × Voxel Volume)

where the summation is performed over the given domain of interest, such as the LV.

By utilizing the viscous dissipation function Φ and the Navier-Stokes energy equations, you can accurately calculate the viscous energy loss in the LV due to frictional forces.

Theoretical Foundations and Formulas

To delve deeper into the theoretical foundations of energy loss due to friction, let’s explore some key physics concepts and formulas:

1. Work-Energy Theorem:
   The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. This principle forms the basis for calculating energy loss due to friction.

Work Done = Change in Kinetic Energy

1. Frictional Force:
   The frictional force is the force that opposes the relative motion between two surfaces in contact. The frictional force can be calculated using the formula:

Frictional Force = Friction Coefficient × Normal Force

The friction coefficient is a dimensionless quantity that depends on the properties of the surfaces in contact.

1. Viscous Dissipation Function Φ:
   The viscous dissipation function Φ in the Newtonian Navier-Stokes energy equations represents the rate of viscous energy dissipation per unit volume. It is defined as:

Φ = μ × [(∂u/∂x)^2 + (∂v/∂y)^2 + (∂w/∂z)^2 + 1/2 × ((∂u/∂y) + (∂v/∂x))^2 + 1/2 × ((∂u/∂z) + (∂w/∂x))^2 + 1/2 × ((∂v/∂z) + (∂w/∂y))^2]

where μ is the dynamic viscosity of the fluid, and u, v, and w are the velocity components in the x, y, and z directions, respectively.

Numerical Examples and Problem-Solving

To solidify your understanding, let’s explore some numerical examples and problem-solving scenarios related to energy loss due to friction:

Example 1: Sliding Block on an Inclined Plane

A 10 kg block is sliding down a 30-degree inclined plane with an initial velocity of 5 m/s. The coefficient of kinetic friction between the block and the plane is 0.2. Calculate the energy lost due to friction over a distance of 5 meters.

Given:
– Mass of the block: 10 kg
– Initial velocity: 5 m/s
– Angle of the inclined plane: 30 degrees
– Coefficient of kinetic friction: 0.2
– Distance traveled: 5 meters

Step 1: Calculate the normal force acting on the block.
Normal Force = Mass × g × cos(θ)
Normal Force = 10 kg × 9.8 m/s^2 × cos(30°) = 86.03 N

Step 2: Calculate the frictional force acting on the block.
Frictional Force = Coefficient of Kinetic Friction × Normal Force
Frictional Force = 0.2 × 86.03 N = 17.21 N

Step 3: Calculate the energy lost due to friction.
Energy Lost Due to Friction = Frictional Force × Distance
Energy Lost Due to Friction = 17.21 N × 5 m = 86.05 J

Example 2: Viscous Energy Loss in the Left Ventricle (LV)

Consider the blood flow in the left ventricle (LV) of the heart. The viscous energy loss can be calculated using the viscous dissipation function Φ in the Newtonian Navier-Stokes energy equations. Assume the following parameters:

• Dynamic viscosity of blood (μ): 0.003 Pa·s
• Velocity gradients in the LV:
• ∂u/∂x = 10 s^-1
• ∂v/∂y = 8 s^-1
• ∂w/∂z = 6 s^-1
• ∂u/∂y + ∂v/∂x = 4 s^-1
• ∂u/∂z + ∂w/∂x = 3 s^-1
• ∂v/∂z + ∂w/∂y = 2 s^-1
• Voxel volume in the LV: 1 mm^3

Calculate the viscous energy loss in the LV over a given time period.

Step 1: Calculate the viscous dissipation function Φ.
Φ = μ × [(∂u/∂x)^2 + (∂v/∂y)^2 + (∂w/∂z)^2 + 1/2 × ((∂u/∂y) + (∂v/∂x))^2 + 1/2 × ((∂u/∂z) + (∂w/∂x))^2 + 1/2 × ((∂v/∂z) + (∂w/∂y))^2]
Φ = 0.003 Pa·s × [(10 s^-1)^2 + (8 s^-1)^2 + (6 s^-1)^2 + 1/2 × (4 s^-1)^2 + 1/2 × (3 s^-1)^2 + 1/2 × (2 s^-1)^2]
Φ = 0.003 Pa·s × (100 + 64 + 36 + 8 + 4.5 + 2) s^-2
Φ = 0.003 Pa·s × 214.5 s^-2
Φ = 0.6435 W/m^3

Step 2: Calculate the viscous energy loss in the LV over a given time period.
Assume the time period is 1 second.
Energy Loss (EL) = Σ (Φ × Voxel Volume)
EL = 0.6435 W/m^3 × 1 mm^3 = 0.6435 mJ

This example demonstrates how to utilize the viscous dissipation function Φ and the Navier-Stokes energy equations to calculate the viscous energy loss in the left ventricle of the heart.

Remember, these are just a few examples, and you can find many more numerical problems and scenarios related to energy loss due to friction in various physics textbooks and online resources.

Conclusion

In this comprehensive guide, we have explored the fundamental principles, formulas, and techniques for calculating the energy lost due to friction. From the basic work-energy theorem to the more complex viscous energy loss in the left ventricle of the heart, we have covered a wide range of concepts and examples to help you develop a deep understanding of this crucial topic in physics.

By mastering the methods and techniques presented in this guide, you will be well-equipped to tackle a variety of problems and scenarios related to energy loss due to friction, both in academic settings and real-world applications. Remember to practice regularly, explore additional resources, and continuously challenge yourself to deepen your knowledge and problem-solving skills.

References

1. Lab#6 – Physics 1AL Introduction ENERGY & ENERGY… https://www.coursehero.com/file/5580043/Lab6/
2. How do you calculate energy loss due to friction? – CK-12 https://www.ck12.org/flexi/life-science/energy-flow/how-do-you-calculate-energy-loss-due-to-friction/
3. What is the mechanical energy lost due to friction? – Physics – Vaia https://www.vaia.com/en-us/textbooks/physics/physics-for-scientists-and-engineers-with-modern-physics-8-edition/chapter-5/problem-44-a-250-kg-child-on-a-200-m-long-swing-is-released-/
4. Assessment of viscous energy loss and the association with three … https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5297883/
5. How to find energy lost (heat or thermal) due to friction? – YouTube https://www.youtube.com/watch?v=yS5mwGwbG9A