We can calculate the angular speed of the conical pendulum using the following steps:

Step 1: Variables and Definitions

Given:

• Mass of the ball, $m = 500 \, \text{g} = 0.5 \, \text{kg}$
• Radius of the circular motion, $r = 20 \, \text{cm} = 0.2 \, \text{m}$
• Length of the string, $L = 1.0 \, \text{m}$

Step 2: Geometry of the Conical Pendulum

For a conical pendulum, the string makes an angle $\theta$ with the vertical. This angle can be determined from the relation: $\sin \theta = \frac{r}{L}$ Substituting the values of $r$ and $L$: $\sin \theta = \frac{0.2}{1.0} = 0.2$ Thus, $\theta = \arcsin(0.2)$.

Step 3: Dynamics of the Conical Pendulum

The forces acting on the pendulum are:

• Tension in the string, $T$
• Gravitational force, $mg$

In equilibrium, the vertical component of tension balances the gravitational force: $T \cos \theta = mg$ The horizontal component provides the centripetal force: $T \sin \theta = m \frac{v^2}{r}$ Where $v$ is the linear speed of the ball. Using these two equations, we can eliminate $T$ and solve for $v$.

Dividing the second equation by the first: $\tan \theta = \frac{v^2}{rg}$ Thus, the linear speed $v$ is: $v = \sqrt{r g \tan \theta}$ Substitute $r = 0.2 \, \text{m}$, $g = 9.8 \, \text{m/s}^2$, and $\tan \theta = 0.2027$ (from $\theta = \arcsin(0.2)$): $v = \sqrt{0.2 \times 9.8 \times 0.2027} \approx 0.632 \, \text{m/s}$

Step 4: Angular Speed Calculation

The angular speed $\omega$ is related to the linear speed $v$ by: $\omega = \frac{v}{r}$ Substituting $v = 0.632 \, \text{m/s}$ and $r = 0.2 \, \text{m}$: $\omega = \frac{0.632}{0.2} \approx 3.16 \, \text{rad/s}$

Step 5: Conversion to RPM

To convert $\omega$ from radians per second to revolutions per minute (rpm), use the following conversion: $1 \, \text{rad/s} = \frac{60}{2\pi} \, \text{rpm}$ Thus, $\omega \approx 3.16 \times \frac{60}{2\pi} \approx 30.2 \, \text{rpm}$

Final Answer:

The angular speed of the ball is approximately 30.2 rpm.

Would you like further clarification or details on any part of the process? Here are five related questions for exploration:

1. How does the length of the string affect the angular speed?
2. What would happen if the mass of the ball was increased while keeping the radius the same?
3. How would the angular speed change if the pendulum was swinging at a larger radius?
4. Can you derive the time period of a conical pendulum?
5. How is the angle $\theta$ affected by changes in the radius of the motion?

Tip: Remember that angular speed depends on both the radius and the gravitational force, so any change in these factors will alter the pendulum's dynamics.