The initial value problem for the damped pendulum equation is
$$\frac{d^2\theta}{dt^2}+k\frac{d\theta}{dt}+\frac{g}{l}\sin{\theta}=0, \quad \theta(0)=\theta_0, \quad \theta'(0)=\omega_0$$

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(b) Non-dimensionalize the model with a time scale appropriate to expecting damping to have a small contribution.
 what is the dimension of theta

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.

The initial value problem for the damped pendulum equation is $$\frac{d^2\theta}{dt^2}+k\frac{d\theta}{dt}+\frac{g}{l}\sin{\theta}=0, \quad \theta(0)=\theta_0, \quad \theta'(0)=\omega_0$$ (b) Non-dimensionalize the model with a time scale appropriate to expecting damping to have a small contribution. What is the dimension of theta?

Explore how to non-dimensionalize the damped pendulum equation with small damping effects. Understand the dimensional properties of angular displacement and apply differential equations to analyze pendulum motion. Suitable for undergraduate physics and mathematics students.

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