Rotation Parametric Equations

Oct 13, 2025 by windailys.com 125 views

Introduction

In mathematics, particularly in calculus and analysis, understanding the motion of objects and calculating the regions they sweep is crucial. This article will delve into the problem of a rotating and moving stick, providing a step-by-step guide on how to fill and calculate the region it sweeps. We will assume a stick with one end fixed at point A(0,0) and the other end at point B(0,1). The stick will rotate counterclockwise around point A with an angular frequency of 2*Pi rad/s while A is moving from (0,0) to (t,0).

Problem Description

To solve this problem, we need to consider the stick's motion as a combination of rotation and translation. The stick's position can be described using parametric equations, which will help us determine the region it sweeps. We will use the following variables:

t: The time parameter, representing the stick's position along the x-axis.
θ: The angular parameter, representing the stick's rotation angle around point A.
x(t): The x-coordinate of point B as a function of time t.
y(t): The y-coordinate of point B as a function of time t.

Parametric Equations

The stick's position can be described using the following parametric equations:

x(t) = t
y(t) = 0

These equations represent the stick's translation along the x-axis.

Rotation Parametric Equations

To describe the stick's rotation around point A, we need to use the following parametric equations:

x(θ) = t + r \* cos(θ)
y(θ) = r \* sin(θ)

where r is the distance between points A and B, which is equal to 1.

Region Swept by the Stick

The region swept by the stick can be described as the area enclosed by the curve traced by point B as the stick rotates and translates. To calculate this area, we need to integrate the area of the infinitesimal regions swept by the stick.

Calculating the Area

To calculate the area swept by the stick, we need to use the following formula:

A = \int_{0}^{2\pi} \int_{0}^{t} r \* dθ \* dx

where r is the distance between points A and B, which is equal to 1.

Evaluating the Integral

To evaluate the integral, we need to substitute the parametric equations into the formula:

A = \int_{0}^{2\pi} \int_{0}^{t} 1 \* dθ \* dx

Evaluating the inner integral, we get:

A = \int_{0}^{2\pi} t \* dθ

Evaluating the outer integral, we get:

A = 2\pi t

Conclusion

In this article, we have shown how to fill and calculate the region swept by a rotating and moving stick. We have used parametric equations to describe the stick's motion and integrated the area of the infinitesimal regions swept by the stick to calculate the total area. The result is a formula for the area swept by the stick, which depends on the time parameter t.

Future Work

This problem can be extended to more complex scenarios, such as a stick with multiple joints or a stick with a varying angular frequency. Additionally, the problem can be generalized to higher dimensions, such as a rotating and moving curve in 3D space.

References

[1] Calculus: Early Transcendentals, James Stewart, 8th edition.
[2] Mathematical Methods for Physics and Engineering, K. F. Riley, M. P. Hobson, and S. J. Bence, 3rd edition.

Code

The following code can be used to visualize the stick's motion and calculate the area swept by the stick:

import numpy as np
import matplotlib.pyplot as plt
def x(t, theta):
return t + np.cos(theta)
def y(t, theta):
return np.sin(theta)
t = np.linspace(0, 10, 100)
theta = np.linspace(0, 2*np.pi, 100)
T, Theta = np.meshgrid(t, theta)
X = x(T, Theta)
Y = y(T, Theta)
plt.plot(X, Y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Stick&#39;s Motion')
plt.show()
A = 2np.pit
print('Area swept by the stick:', A)

This code uses the parametric equations to calculate the x and y coordinates of the stick's motion and plots the stick's motion using matplotlib. It also calculates the area swept by the stick using the formula derived in this article.

Introduction

In our previous article, we discussed how to fill and calculate the region swept by a rotating and moving stick. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the main difference between a rotating and moving stick and a static stick?

A: The main difference between a rotating and moving stick and a static stick is that the rotating and moving stick has a changing position and orientation over time, while a static stick has a fixed position and orientation.

Q: How do I determine the parametric equations for a rotating and moving stick?

A: To determine the parametric equations for a rotating and moving stick, you need to consider the stick's motion as a combination of rotation and translation. You can use the following variables:

t: The time parameter, representing the stick's position along the x-axis.
θ: The angular parameter, representing the stick's rotation angle around the fixed point.
x(t): The x-coordinate of the stick's end point as a function of time t.
y(t): The y-coordinate of the stick's end point as a function of time t.

Q: How do I calculate the area swept by a rotating and moving stick?

A: To calculate the area swept by a rotating and moving stick, you need to integrate the area of the infinitesimal regions swept by the stick. You can use the following formula:

A = \int_{0}^{2\pi} \int_{0}^{t} r \* dθ \* dx

where r is the distance between the fixed point and the stick's end point.

Q: What is the significance of the angular frequency in a rotating and moving stick?

A: The angular frequency is the rate at which the stick rotates around the fixed point. It is an important parameter in determining the stick's motion and the area swept by the stick.

Q: Can I use the same method to calculate the area swept by a rotating and moving curve in 3D space?

A: Yes, you can use the same method to calculate the area swept by a rotating and moving curve in 3D space. However, you need to consider the additional dimension and use the appropriate parametric equations to describe the curve's motion.

Q: How do I visualize the stick's motion and the area swept by the stick?

A: You can use computer-aided design (CAD) software or programming languages like Python or MATLAB to visualize the stick's motion and the area swept by the stick. You can also use graphing tools like matplotlib or plotly to create interactive visualizations.

Q: What are some real-world applications of calculating the area swept by a rotating and moving stick?

A: Calculating the area swept by a rotating and moving stick has many real-world applications, such as:

Robotics: Calculating the area swept by a rotating and moving stick is essential in robotics to determine the workspace of a robotic arm or a robotic gripper.
Mechanical Engineering: Calculating the area swept by a rotating and moving stick is important in mechanical engineering to determine the motion of a mechanical system and the forces acting on it.
Computer-Aided Design (CAD): Calculating the area swept by a rotating and moving stick is useful in CAD to create complex shapes and models.

Conclusion

In this article, we have answered some frequently asked questions related to calculating the area swept by a rotating and moving stick. We hope that this article has provided you with a better understanding of the topic and has helped you to apply the concepts to real-world problems.

Future Work

This topic can be extended to more complex scenarios, such as a stick with multiple joints or a stick with a varying angular frequency. Additionally, the problem can be generalized to higher dimensions, such as a rotating and moving curve in 3D space.

References

[1] Calculus: Early Transcendentals, James Stewart, 8th edition.
[2] Mathematical Methods for Physics and Engineering, K. F. Riley, M. P. Hobson, and S. J. Bence, 3rd edition.

Code

The following code can be used to visualize the stick's motion and calculate the area swept by the stick:

import numpy as np
import matplotlib.pyplot as plt
def x(t, theta):
return t + np.cos(theta)
def y(t, theta):
return np.sin(theta)
t = np.linspace(0, 10, 100)
theta = np.linspace(0, 2*np.pi, 100)
T, Theta = np.meshgrid(t, theta)
X = x(T, Theta)
Y = y(T, Theta)
plt.plot(X, Y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Stick&#39;s Motion')
plt.show()
A = 2np.pit
print('Area swept by the stick:', A)