Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**The Restoring Force of a Simple Pendulum**
The restoring force in the angular (\(\theta\)) direction for a simple pendulum is given by \(F_{\theta} = -mg \sin \theta\). The corresponding acceleration is \(a_{\theta} = -F_{\theta}/m = -g \sin \theta\). Dividing \(a_{\theta}\) by the pendulum length \(L\) yields the angular acceleration:
\[
\alpha = a_{\theta}/L = -\frac{g}{L} \sin \theta
\]
From rotational kinematics, we know that angular acceleration is the rate of change of angular velocity, i.e., \(\alpha = \frac{d \Omega}{dt} = \dot{\Omega}\), and the angular velocity is the rate of change of the angle \(\theta\), i.e.,
\[
\Omega = \frac{d \theta}{dt} = \dot{\theta}
\]
Thus, the differential equations of motion for the simple pendulum are:
\[
\frac{d \Omega}{dt} = -\frac{g}{L} \sin \theta
\]
\[
\frac{d \theta}{dt} = \Omega
\]
*Why do we use a capital Greek omega (\(\Omega\)) for angular velocity? Because the lowercase \(\omega\) is defined in the lab manual as the angular frequency of the pendulum, which is a different (constant) quantity:*
\[
\omega = \sqrt{\frac{g}{L}}
\]
---
**Iteration is King (again)! The Euler-Cromer Method of Numerical Integration**
To solve these differential equations, we use a method similar to the interactive Euler method as for the "free fall/air drag" experiment. First, let's define the initial values: We start the pendulum at time \(t_0\) from an angle \(\theta_0\) with zero angular velocity (\(\Omega_0 = 0\)).
For a sufficiently short time interval \(\Delta t\), the angular acceleration can be considered constant, allowing us to calculate the new angular velocity at time \(t_1 = t_0 + \Delta t\):
\[
\Omega_1 = \Omega_0 + \frac{d \Omega}{dt} \Delta t = \Omega_0 - \frac{g}{L](https://content.bartleby.com/qna-images/question/c303f822-5e20-4d30-8920-178170554c96/b22dbd31-60a6-4c82-8ea7-6426beac0fb6/alegz_processed.png)
Transcribed Image Text:**The Restoring Force of a Simple Pendulum**
The restoring force in the angular (\(\theta\)) direction for a simple pendulum is given by \(F_{\theta} = -mg \sin \theta\). The corresponding acceleration is \(a_{\theta} = -F_{\theta}/m = -g \sin \theta\). Dividing \(a_{\theta}\) by the pendulum length \(L\) yields the angular acceleration:
\[
\alpha = a_{\theta}/L = -\frac{g}{L} \sin \theta
\]
From rotational kinematics, we know that angular acceleration is the rate of change of angular velocity, i.e., \(\alpha = \frac{d \Omega}{dt} = \dot{\Omega}\), and the angular velocity is the rate of change of the angle \(\theta\), i.e.,
\[
\Omega = \frac{d \theta}{dt} = \dot{\theta}
\]
Thus, the differential equations of motion for the simple pendulum are:
\[
\frac{d \Omega}{dt} = -\frac{g}{L} \sin \theta
\]
\[
\frac{d \theta}{dt} = \Omega
\]
*Why do we use a capital Greek omega (\(\Omega\)) for angular velocity? Because the lowercase \(\omega\) is defined in the lab manual as the angular frequency of the pendulum, which is a different (constant) quantity:*
\[
\omega = \sqrt{\frac{g}{L}}
\]
---
**Iteration is King (again)! The Euler-Cromer Method of Numerical Integration**
To solve these differential equations, we use a method similar to the interactive Euler method as for the "free fall/air drag" experiment. First, let's define the initial values: We start the pendulum at time \(t_0\) from an angle \(\theta_0\) with zero angular velocity (\(\Omega_0 = 0\)).
For a sufficiently short time interval \(\Delta t\), the angular acceleration can be considered constant, allowing us to calculate the new angular velocity at time \(t_1 = t_0 + \Delta t\):
\[
\Omega_1 = \Omega_0 + \frac{d \Omega}{dt} \Delta t = \Omega_0 - \frac{g}{L
![**Exercise 1: Complete the Euler-Cromer Loop Code Below**
Write the code for calculating \( t_i \), \( \Omega_i \), and \( \theta_i \) based on the equations above. As before, we use a `for` loop for the repetitive (iterative) calculations. Remember that only indented lines become part of the loop, and all those lines must have the same indentation.
- The `sin` function comes from the `numpy` library, so call it as `np.sin()`.
- For the time step, use the variable `dt` defined earlier.
- The \( i \)-th elements of time, angle, and angular velocity can be accessed as `t[i]`, `theta[i]`, and `Omega[i]`, respectively. For element number \( i - 1 \), replace \( i \) with \( i-1 \).
```python
# Initial conditions. Vector numbering in Python starts at 0.
time[0] = 0 # initial time
theta[0] = np.deg2rad(5.0) # initial angle, converted to radians.
Omega[0] = 0 # initial angular velocity (pendulum starts at rest)
# Main loop: Euler algorithm and evaluation of exact solutions for v and y
#
# Your code should look like this (without the # signs, of course):
# for i in range(1, t_steps):
# time[i] = ....
# Omega[i] = ....
# theta[i] = ....
# YOUR CODE HERE
raise NotImplementedError()
```
**Explanation:**
This code snippet outlines the initial setup and structure for implementing the Euler-Cromer method to simulate a physical system, such as a pendulum. The initial conditions are provided, and users are instructed to fill in the iterative loop for calculating the time, angular velocity, and angular position at each step.](https://content.bartleby.com/qna-images/question/c303f822-5e20-4d30-8920-178170554c96/b22dbd31-60a6-4c82-8ea7-6426beac0fb6/lp4het_processed.png)
Transcribed Image Text:**Exercise 1: Complete the Euler-Cromer Loop Code Below**
Write the code for calculating \( t_i \), \( \Omega_i \), and \( \theta_i \) based on the equations above. As before, we use a `for` loop for the repetitive (iterative) calculations. Remember that only indented lines become part of the loop, and all those lines must have the same indentation.
- The `sin` function comes from the `numpy` library, so call it as `np.sin()`.
- For the time step, use the variable `dt` defined earlier.
- The \( i \)-th elements of time, angle, and angular velocity can be accessed as `t[i]`, `theta[i]`, and `Omega[i]`, respectively. For element number \( i - 1 \), replace \( i \) with \( i-1 \).
```python
# Initial conditions. Vector numbering in Python starts at 0.
time[0] = 0 # initial time
theta[0] = np.deg2rad(5.0) # initial angle, converted to radians.
Omega[0] = 0 # initial angular velocity (pendulum starts at rest)
# Main loop: Euler algorithm and evaluation of exact solutions for v and y
#
# Your code should look like this (without the # signs, of course):
# for i in range(1, t_steps):
# time[i] = ....
# Omega[i] = ....
# theta[i] = ....
# YOUR CODE HERE
raise NotImplementedError()
```
**Explanation:**
This code snippet outlines the initial setup and structure for implementing the Euler-Cromer method to simulate a physical system, such as a pendulum. The initial conditions are provided, and users are instructed to fill in the iterative loop for calculating the time, angular velocity, and angular position at each step.
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