Question

Consider the motion of a simple pendulum governed by the equation $$ \frac{d \omega}{d \theta}=-\frac{g}{l} \frac{\sin \theta}{\omega} $$ where \(\omega\) is the angular velocity in radians per second, \(\theta\) is the angle of swing in degrees, \(g=32.2014\) feet per second per second, and \(l=2\) feet. If the initial condition is \(\omega(\pi / 3)=\frac{1}{2}\), find \(\omega\left(70^{\circ}\right)\) by applying the Runge-Kutta method with Runge's coefficients. Use \(\Delta \theta=0.5^{\circ}\).

Short answer

Answer: To find the angular velocity of the pendulum at an angle of \(70^{\circ}\), perform the following steps: 1. Understand the Runge-Kutta method and its implementation. 2. Define the governing equation as a function. 3. Set up initial conditions and step size. 4. Implement the Runge-Kutta method with Runge's coefficients. 5. Calculate the angular velocity at \(70^{\circ}\). After performing these steps, you will find the desired angular velocity at an angle of \(70^{\circ}\).

Step-by-step solution

Analyze the Runge-Kutta method and its implementation

The Runge-Kutta method is an iterative method commonly used to solve ordinary differential equations (ODEs) by approximating the solution at discrete points. In this case, we have an ODE relating the angular velocity (\(\omega\)) and the angle of swing (\(\theta\)), so we can use the Runge-Kutta method to approximate \(\omega\) at the desired angle. To implement the Runge-Kutta method with Runge's coefficients, we will use the following formulas for second-order Runge-Kutta method (also known as Heun's method): $$ k_{1} = h\cdot f\left(\theta, \omega(\theta)\right) $$ $$ k_{2} = h\cdot f\left(\theta+h, \omega(\theta)+k_{1}\right) $$ $$ \omega(\theta+h) = \omega(\theta) + \frac{1}{2}\left(k_{1}+k_{2}\right) $$ Where \(h\) is the step size, \(\theta\) is the current angle of swing, and \(f(\theta, \omega)\) is our governing equation.

Define the governing equation as a function

We will represent the given equation in a function form to implement the Runge-Kutta method: $$ f\left(\theta, \omega\right) = -\frac{g}{l} \frac{\sin \theta}{\omega} $$ Here, we have \(g = 32.2014\) ft/s² and \(l = 2\) ft.

Set up initial conditions and step size

We are given the initial condition: \(\omega\left(\frac{\pi}{3}\right) = \frac{1}{2}\). We need to find the omega value at \(70^{\circ}\). We are also given the step size \(\Delta \theta = 0.5^{\circ}\). We will first convert all the angles to radians for calculations. Initial angle: \(\theta_0 = \frac{\pi}{3}\) radians Final angle: \(\theta_f = 70^{\circ} \cdot \frac{\pi}{180} = \frac{7\pi}{18}\) radians Step size: \(h = 0.5^{\circ} \cdot \frac{\pi}{180} = \frac{\pi}{360}\) radians

Implement the Runge-Kutta method with Runge's coefficients

Now we will apply the Runge-Kutta method to find the value of \(\omega\left(70^{\circ}\right)\). Using the formulas defined earlier, we will iterate through the angles until we reach the desired angle, updating the value of \(\omega\) at each step.

Calculate the angular velocity at \(70^{\circ}\)

After implementing the Runge-Kutta method, we will find the value of \(\omega\left(\frac{7\pi}{18}\right)\), which is the angular velocity of the pendulum at \(70^{\circ}\). Once the iterations are complete, the final value of \(\omega\) will be the desired angular velocity at \(70^{\circ}\). In summary, we first understood the Runge-Kutta method and its implementation. Then, we applied the method with the given initial conditions and step size to find the angular velocity at the desired angle of swing, \(70^{\circ}\).

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that involve functions of only one independent variable and their derivatives. They play a crucial role in many areas of science and engineering. ODEs describe how quantities change and are used for modeling a variety of dynamic systems, such as population growth, chemical reactions, and physical systems.

A key feature of ODEs is that they express a relationship between a function and its rates of change. In our context, the motion of a simple pendulum is described by an ODE that relates the angular velocity \( \omega \) with the angle \( \theta \).

• The general form of an ODE is \( F(x, y, y') = 0 \), where \( y \) is a function of \( x \), and \( y' \) is its derivative.
• In the pendulum problem, \( \omega \) is equivalent to \( y \), and \( \theta \) is equivalent to \( x \).

Understanding and solving ODEs is fundamental for predicting system behaviors, such as the time evolution of a pendulum's motion. Numerical methods like Runge-Kutta are often used to find approximate solutions where analytical solutions are difficult to obtain.

Simple Pendulum Motion

The motion of a simple pendulum over time can be described by considering its angular position and velocity. A simple pendulum consists of a weight suspended from a pivot, allowing it to swing freely under the influence of gravity.

For small angles, the motion can be approximated by a simple harmonic oscillator. However, for larger angles, the governing ODE becomes non-linear, as seen with the inclusion of the sine function. Useful details about simple pendulum motion include:

• The equation \( \frac{d \omega}{d \theta} = -\frac{g}{l} \frac{\sin \theta}{\omega} \) defines the relationship between angular velocity \( \omega \) and the angle \( \theta \).
• The variable \( g \) represents the acceleration due to gravity, and \( l \) is the length of the pendulum.
• This configuration represents a common physical system and provides an excellent basis for studying differential equations.

By understanding these concepts, we gain insights into how the pendulum's angle and angular velocity are interconnected through time.

Numerical Methods

In situations where analytical solutions of differential equations are challenging, numerical methods come into play. These methods approximate solutions using computational algorithms. The Runge-Kutta method is one such widely used technique for solving ODEs. It helps in iteratively calculating the trajectory of a system, step by step.

A few key aspects of numerical methods like Runge-Kutta include:

• It's applicable to a wide range of problems, enabling detailed analysis of complex systems.
• The second-order Runge-Kutta, also known as Heun's method, is an example that balances accuracy with computational efficiency.
• It essentially predicts the state of a system by considering the slope at several points within each step interval.

In our pendulum exercise, these methods allow us to estimate angular velocity as it swings to a specific angle \(70^{\circ}\), given initial conditions and a step size. Through each iteration, the method refines the solution, highlighting its efficiency and accuracy in practical applications.

Initial Conditions

Initial conditions are the starting values that set the stage for solving differential equations. They specify the state of a system at the beginning of the analysis. They are crucial, as they influence the entire trajectory of a solution.

In the context of our exercise, an initial condition is given as \( \omega(\frac{\pi}{3})=\frac{1}{2} \).

• This condition signifies the angular velocity when the pendulum's angle is \( 60^{\circ} \).
• Initial conditions are vital for numerically calculating future states using methods like Runge-Kutta. They anchor the calculations, providing a precise starting point.
• By combining these conditions with the mathematical model, we can simulate the pendulum's evolution to reach the desired outcome.

The defined initial state ensures that subsequent calculations are built on a solid foundation, enabling accurate predictions of the pendulum's behavior.