To non-dimensionalize the given damped pendulum equation, let's proceed step by step and explore the dimension of $\theta$.

Step 1: Rewrite the equation

The given equation for the damped pendulum is:

$\frac{d^2 \theta}{dt^2} + k \frac{d\theta}{dt} + \frac{g}{l} \sin{\theta} = 0,$ with initial conditions: $\theta(0) = \theta_0, \quad \theta'(0) = \omega_0.$

Here:

• $\theta(t)$ is the angular displacement.
• $k$ is the damping constant (units of inverse time).
• $g$ is the gravitational acceleration (units of $\text{m/s}^2$).
• $l$ is the length of the pendulum (units of meters).
• $\theta_0$ and $\omega_0$ are initial angular displacement and velocity, respectively.

Step 2: Determine the dimension of $\theta$

The angle $\theta$ represents the angular displacement of the pendulum, which is a measure of rotation. Since $\theta$ is an angular variable, its dimension is:

$\boxed{\text{Dimension of } \theta = 1.}$

In other words, $\theta$ is dimensionless because angles are measured in radians (which are dimensionless quantities in the SI unit system). A radian is defined as the ratio of the length of the arc to the radius, both of which have the same units, making their ratio dimensionless.

Would you like a detailed explanation on how to non-dimensionalize the whole model?

Here are 5 related questions:

1. What is the time scale used for non-dimensionalization of this model?
2. How does small damping affect the motion of the pendulum over time?
3. What are the physical interpretations of $k$ and $\frac{g}{l}$ in the pendulum equation?
4. How do we transform the equation after non-dimensionalization?
5. What happens to the frequency of oscillations when damping is small?

Tip: Non-dimensionalization helps simplify a model by reducing the number of parameters, making it easier to understand key physical behaviors.