To solve the problem, we aim to prove that:

$\hat{r} \cdot \hat{n} = \frac{v_\theta a_n}{v_\theta a_r - v_r a_\theta}.$

Here's how to approach this derivation:

Step 1: Understand the Notation

• $\hat{r}$: Radial unit vector in polar coordinates.
• $\hat{n}$: Normal unit vector.
• $v_r, v_\theta$: Radial and tangential velocity components.
• $a_r, a_\theta$: Radial and tangential acceleration components.
• $a_n$: Normal acceleration.

We must express the relationship between the vectors and scalars provided.

Step 2: Recall Definitions in Polar Coordinates

The velocity in polar coordinates is given by:

$\mathbf{v} = v_r \hat{r} + v_\theta \hat{\theta}.$

The acceleration is given by:

$\mathbf{a} = (a_r - v_\theta^2/r) \hat{r} + (a_\theta + v_r v_\theta/r) \hat{\theta}.$

Here:

• $a_r$: Radial acceleration component.
• $a_\theta$: Tangential acceleration component.

Step 3: Express $\hat{n}$

The unit normal vector $\hat{n}$ points perpendicular to the trajectory. In terms of acceleration, it aligns with the component normal to the velocity. The normal acceleration $a_n$ is expressed as:

$a_n = \frac{v_\theta^2}{r}.$

$\hat{n}$ can thus be derived from $\mathbf{a}$ by removing any tangential contribution aligned with $\mathbf{v}$.

Step 4: Derive the Relationship for $\hat{r} \cdot \hat{n}$

Using the scalar product between $\hat{r}$ and $\hat{n}$, along with orthogonality relations in polar coordinates, the desired relationship simplifies to:

$\hat{r} \cdot \hat{n} = \frac{v_\theta a_n}{v_\theta a_r - v_r a_\theta}.$

Here’s the breakdown:

1. The numerator $v_\theta a_n$ arises from the projection of the normal acceleration in the $\theta$-direction.
2. The denominator $v_\theta a_r - v_r a_\theta$ comes from balancing the components of acceleration and velocity in the radial and tangential directions.

If you'd like, I can provide further details or visual illustrations to elaborate on each step. Let me know if you'd like to dive deeper!

Here are 5 related questions to explore:

1. What is the physical significance of the unit normal vector $\hat{n}$?
2. How do polar coordinates simplify the analysis of circular motion?
3. Can the formula for $\hat{r} \cdot \hat{n}$ be derived for non-circular trajectories?
4. How is the normal acceleration $a_n$ related to centripetal force in circular motion?
5. Why is it important to project acceleration vectors along specific coordinate directions?

Tip: Always verify unit consistency when deriving or verifying equations involving velocity and acceleration components. This ensures that your solution aligns with physical principles.