To find the angular speed $\omega$ of the ball in a conical pendulum, we need to analyze the forces acting on the ball and relate them to the motion in circular motion.

Step-by-Step Derivation:

1. Forces Acting on the Ball:

   • The tension $T$ in the string has two components:
     • The vertical component, $T \cos \theta$, balances the weight of the ball $mg$.
     • The horizontal component, $T \sin \theta$, provides the centripetal force for the circular motion.

2. Trigonometry in the Conical Pendulum: The ball moves in a horizontal circle with radius $r$, and the string makes an angle $\theta$ with the vertical. Using geometry, we have:

   \quad \text{and} \quad \sin \theta = \frac{r}{L}$$

3. Vertical Force Balance: The vertical component of the tension balances the gravitational force: $T \cos \theta = mg$

4. Horizontal Force (Centripetal Force): The horizontal component of the tension provides the centripetal force necessary for circular motion: $T \sin \theta = m \frac{v^2}{r}$ where $v$ is the linear speed of the ball.

5. Substitute for $T$: From the vertical force equation, solve for $T$: $T = \frac{mg}{\cos \theta}$ Substituting this into the horizontal force equation: $\frac{mg}{\cos \theta} \sin \theta = m \frac{v^2}{r}$ Simplifying: $g \tan \theta = \frac{v^2}{r}$

6. Angular Speed $\omega$: The angular speed $\omega$ is related to the linear speed $v$ by $v = \omega r$. Substituting $v = \omega r$ into the previous equation: $g \tan \theta = \omega^2 r$

7. Relating $\tan \theta$ to $r$ and $L$: From the geometry of the pendulum: $\tan \theta = \frac{r}{\sqrt{L^2 - r^2}}$ Substituting this into the equation $g \tan \theta = \omega^2 r$: $g \frac{r}{\sqrt{L^2 - r^2}} = \omega^2 r$ Simplifying by dividing both sides by $r$ (assuming $r \neq 0$): $g \frac{1}{\sqrt{L^2 - r^2}} = \omega^2$

8. Final Expression for $\omega$: Solving for $\omega$: $\omega = \sqrt{\frac{g}{\sqrt{L^2 - r^2}}}$

Thus, the angular speed $\omega$ of the ball is: $\boxed{\omega = \sqrt{\frac{g}{\sqrt{L^2 - r^2}}}}$

Do you need any details or have any questions?

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Here are 5 related questions:

1. How would the angular speed $\omega$ change if the radius $r$ increased while keeping $L$ constant?
2. What would happen to $\omega$ if the length of the string $L$ increased?
3. How does the mass of the ball affect the angular speed in this conical pendulum?
4. Can you derive an expression for the tension in the string in terms of $L$, $r$, and $g$?
5. How would the conical pendulum behave if the system were placed on a planet with a different gravitational acceleration $g$?

Tip: In conical pendulum problems, always check the geometric relationships between the string length, radius, and the angle the string makes with the vertical to simplify your calculations.