A conical pendulum is formed by attaching a ball of mass m to a string of length L, then allowing the ball to move in a horizontal circle of radius r. What is angular speed (in  rpm ) or a 500  g  ball swinging in a 20- cm -radius circle at the end of a 1.0- m -long string?

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.

A conical pendulum is formed by attaching a ball of mass m to a string of length L, then allowing the ball to move in a horizontal circle of radius r. What is angular speed (in rpm) for a 500 g ball swinging in a 20-cm-radius circle at the end of a 1.0-m-long string?

Explore how to calculate the angular speed of a 500g ball swinging in a conical pendulum with a radius of 20cm and string length of 1m. Using trigonometry and centripetal force concepts, we find the speed to be approximately 30.2 rpm. Suitable for high school students.

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