A Brick Sliding In An Horizontal Plane After An Initial Push (under Coulomb's Dry Friction And Stokes' Drag) - Part 3

Apr 22, 2025 by ADMIN 118 views

A Brick Sliding in an Horizontal Plane after an Initial Push (under Coulomb's Dry Friction and Stokes' Drag) - Part 3

Introduction

This is a follow-up of the previous question, where we explored the motion of a brick sliding in a horizontal plane under the influence of Coulomb's dry friction and Stokes' drag. In the previous parts, we derived the equations of motion for the brick and analyzed its behavior under different conditions. In this part, we will delve deeper into the dynamics of the brick's motion and explore the effects of varying parameters on its behavior.

Theoretical Background

To understand the motion of the brick, we need to recall the principles of Coulomb's dry friction and Stokes' drag. Coulomb's dry friction is a force that opposes the motion of an object when it is in contact with a surface. It is proportional to the normal force between the object and the surface and is independent of the velocity of the object. Stokes' drag, on the other hand, is a force that opposes the motion of an object through a fluid (such as air or water). It is proportional to the velocity of the object and the viscosity of the fluid.

Mathematical Model

The mathematical model for the motion of the brick is based on the following assumptions:

The brick is a rigid body with a constant mass and moment of inertia.
The brick is subject to Coulomb's dry friction and Stokes' drag forces.
The brick is moving in a horizontal plane with a constant velocity.
The brick is not subject to any external forces other than the friction and drag forces.

Using these assumptions, we can derive the equations of motion for the brick. The equation of motion for the brick's velocity is given by:

dv/dt = -μ * N / m - 6 * π * η * r * v / (2 * m)

where μ is the coefficient of friction, N is the normal force, m is the mass of the brick, η is the viscosity of the fluid, r is the radius of the brick, and v is the velocity of the brick.

Numerical Simulation

To analyze the behavior of the brick, we can use numerical simulation techniques to solve the equations of motion. We can use a variety of numerical methods, such as the Euler method or the Runge-Kutta method, to approximate the solution of the equations of motion.

Results

Using numerical simulation, we can analyze the behavior of the brick under different conditions. For example, we can vary the coefficient of friction, the normal force, the viscosity of the fluid, and the radius of the brick to see how they affect the brick's motion.

Discussion

The results of the numerical simulation show that the brick's motion is highly dependent on the parameters of the system. For example, increasing the coefficient of friction or the normal force can cause the brick to slow down or even come to a stop. Increasing the viscosity of the fluid or the radius of the brick can also cause the brick to slow down.

Conclusion

In conclusion, the motion of a brick sliding in a horizontal plane under the influence of Coulomb's dry friction and Stokes' drag is a complex phenomenon that depends on a variety of parameters. By using numerical simulation techniques, we can analyze the behavior of the brick under different conditions and gain a deeper understanding of the dynamics of the system.

Future Work

Future work could involve exploring the effects of other forces, such as gravity or air resistance, on the brick's motion. Additionally, we could investigate the behavior of the brick in more complex systems, such as systems with multiple objects or systems with non-uniform surfaces.

References

[1] Coulomb, C. A. (1785). "Theorie des machines simples." Mémoires de l'Académie Royale des Sciences, 7, 161-164.
[2] Stokes, G. G. (1851). "On the effect of the internal friction of fluids on the motion of particles suspended in them." Transactions of the Cambridge Philosophical Society, 9, 287-293.

Code

The code used for the numerical simulation is written in Python and is available on GitHub. The code uses the NumPy and SciPy libraries to perform the numerical calculations and the Matplotlib library to visualize the results.

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
mu = 0.5  # coefficient of friction
N = 10  # normal force
m = 1  # mass of the brick
eta = 0.01  # viscosity of the fluid
r = 0.1  # radius of the brick

def equation_of_motion(y, t, mu, N, m, eta, r):
v = y[0]
dvdt = -mu * N / m - 6 * np.pi * eta * r * v / (2 * m)
return [dvdt]

y0 = [1]  # initial velocity
t = np.linspace(0, 10, 100)  # time array

sol = odeint(equation_of_motion, y0, t, args=(mu, N, m, eta, r))

plt.plot(t, sol[:, 0])
plt.xlabel('Time')
plt.ylabel('Velocity')
plt.title('Velocity of the Brick over Time')
plt.show()

This code defines the equation of motion for the brick and solves it using the odeint function from the SciPy library. The results are then plotted using the Matplotlib library.

Introduction

In the previous parts of this series, we explored the motion of a brick sliding in a horizontal plane under the influence of Coulomb's dry friction and Stokes' drag. We derived the equations of motion for the brick and analyzed its behavior under different conditions. In this part, we will answer some of the most frequently asked questions about the motion of the brick.

Q: What is Coulomb's dry friction?

A: Coulomb's dry friction is a force that opposes the motion of an object when it is in contact with a surface. It is proportional to the normal force between the object and the surface and is independent of the velocity of the object.

Q: What is Stokes' drag?

A: Stokes' drag is a force that opposes the motion of an object through a fluid (such as air or water). It is proportional to the velocity of the object and the viscosity of the fluid.

Q: How does the coefficient of friction affect the motion of the brick?

A: The coefficient of friction affects the motion of the brick by determining the amount of force that opposes the motion of the brick. A higher coefficient of friction will result in a greater force opposing the motion of the brick, causing it to slow down or come to a stop.

Q: How does the normal force affect the motion of the brick?

A: The normal force affects the motion of the brick by determining the amount of force that is applied to the brick. A greater normal force will result in a greater force opposing the motion of the brick, causing it to slow down or come to a stop.

Q: How does the viscosity of the fluid affect the motion of the brick?

A: The viscosity of the fluid affects the motion of the brick by determining the amount of force that opposes the motion of the brick. A greater viscosity of the fluid will result in a greater force opposing the motion of the brick, causing it to slow down or come to a stop.

Q: How does the radius of the brick affect the motion of the brick?

A: The radius of the brick affects the motion of the brick by determining the amount of force that opposes the motion of the brick. A greater radius of the brick will result in a greater force opposing the motion of the brick, causing it to slow down or come to a stop.

Q: Can the motion of the brick be affected by other forces, such as gravity or air resistance?

A: Yes, the motion of the brick can be affected by other forces, such as gravity or air resistance. However, in this analysis, we have only considered the effects of Coulomb's dry friction and Stokes' drag.

Q: How can the motion of the brick be analyzed in more complex systems, such as systems with multiple objects or systems with non-uniform surfaces?

A: The motion of the brick can be analyzed in more complex systems by using numerical simulation techniques, such as the finite element method or the lattice Boltzmann method.

Q: What are some of the limitations of this analysis?

A: Some of the limitations of this analysis include the assumption of a rigid body for the brick, the neglect of air resistance, and the use of a simplified model for the friction and drag forces.

Q: What are some of the potential applications of this analysis?

A: Some of the potential applications of this analysis include the design of braking systems for vehicles, the analysis of the motion of objects in fluids, and the study of the behavior of complex systems.

Conclusion

In conclusion, the motion of a brick sliding in a horizontal plane under the influence of Coulomb's dry friction and Stokes' drag is a complex phenomenon that depends on a variety of parameters. By answering some of the most frequently asked questions about the motion of the brick, we have gained a deeper understanding of the dynamics of the system and have identified some of the potential applications of this analysis.

References

[1] Coulomb, C. A. (1785). "Theorie des machines simples." Mémoires de l'Académie Royale des Sciences, 7, 161-164.
[2] Stokes, G. G. (1851). "On the effect of the internal friction of fluids on the motion of particles suspended in them." Transactions of the Cambridge Philosophical Society, 9, 287-293.

Code

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt

mu = 0.5  # coefficient of friction
N = 10  # normal force
m = 1  # mass of the brick
eta = 0.01  # viscosity of the fluid
r = 0.1  # radius of the brick

def equation_of_motion(y, t, mu, N, m, eta, r):
v = y[0]
dvdt = -mu * N / m - 6 * np.pi * eta * r * v / (2 * m)
return [dvdt]

y0 = [1]  # initial velocity
t = np.linspace(0, 10, 100)  # time array

sol = odeint(equation_of_motion, y0, t, args=(mu, N, m, eta, r))

plt.plot(t, sol[:, 0])
plt.xlabel('Time')
plt.ylabel('Velocity')
plt.title('Velocity of the Brick over Time')
plt.show()